Links:

Examples of Stan code for regression models

What is in the stan fit object?

1 Data set Howell

Dataset Howell from the rethinking package.

data(Howell1)
howell1 <- Howell1
head(howell1)
##    height   weight age male
## 1 151.765 47.82561  63    1
## 2 139.700 36.48581  63    0
## 3 136.525 31.86484  65    0
## 4 156.845 53.04191  41    1
## 5 145.415 41.27687  51    0
## 6 163.830 62.99259  35    1
str(howell1)
## 'data.frame':    544 obs. of  4 variables:
##  $ height: num  152 140 137 157 145 ...
##  $ weight: num  47.8 36.5 31.9 53 41.3 ...
##  $ age   : num  63 63 65 41 51 35 32 27 19 54 ...
##  $ male  : int  1 0 0 1 0 1 0 1 0 1 ...
ggplot(howell1, aes(height),bins=50) + geom_histogram(aes(y = ..count..), fill="darkgreen",binwidth = 1, alpha=0.4) + 
  theme(axis.line = element_line(size=1, colour = "black"),
              panel.grid.major = element_line(colour = "#d3d3d3"),
              panel.grid.minor = element_blank())

Subset with only individuals over 18 years old:

howell2 <- howell1[howell1$age >= 18 , ]
plot_grid(ggplot(howell2, aes(height),bins=50) + geom_histogram(aes(y = ..count..), fill="darkgreen",binwidth = 1, alpha=0.4) + 
  theme(axis.line = element_line(size=1, colour = "black"),
              panel.grid.major = element_line(colour = "#d3d3d3"),
              panel.grid.minor = element_blank()),
  ggplot(howell2, aes(weight),bins=50) + geom_histogram(aes(y = ..count..), fill="blue",binwidth = 1, alpha=0.4) + 
  theme(axis.line = element_line(size=1, colour = "black"),
              panel.grid.major = element_line(colour = "#d3d3d3"),
              panel.grid.minor = element_blank()))

ggplot(howell2,aes(y=height,x=weight))+geom_point() + theme(axis.line = element_line(size=1, colour = "black"),
              panel.grid.major = element_line(colour = "gray95"),
              panel.grid.minor = element_blank())

2 Without predictor

2.1 With large variance

\[ h_{i} \sim \mathcal{N}(\mu \, , \, \sigma^2)\] \[\mu \sim \mathcal{N}(178,20)\]

Data in a list:

m1.data <- list(N = nrow(howell2), height = howell2$height)

2.1.1 Notation 1 (vectorized with target += notation)

explicitly incrementing the log probability function

https://mc-stan.org/docs/2_18/reference-manual/increment-log-prob-section.html

“The basis of Stan’s execution is the evaluation of a log probability function (specifically, a probability density function) for a given set of (real-valued) parameters; this function returns the log density of the posterior up to an additive constant. Data and transformed data are fixed before the log density is evaluated. The total log probability is initialized to zero. Next, any log Jacobian adjustments accrued by the variable constraints are added to the log density (the Jacobian adjustment may be skipped for optimization). Sampling and log probability increment statements may add to the log density in the model block. A log probability increment statement directly increments the log density with the value of an expression as follows.”

https://stackoverflow.com/questions/40289457/stan-using-target-syntax

“The target density is the density from which the sampler samples and it needs to be equal to the joint density of all the parameters given the data up to a constant (which is usually achieved via Bayes’s rule by coding as the joint density of parameters and modeled data up to a constant).”

data {                                             // observed variables 
  int<lower=1> N;                                  // number of observations
  vector[N] height;                                // response variables
}
parameters {                                       // unobserved variables
  real mu;                                      
  real<lower=0,upper=50> sigma;
}
model {
  target += normal_lpdf(height | mu, sigma);       // likelihood
  target += normal_lpdf(mu | 178, 20);             // prior of the mean
}
fit.m1.1 <- sampling(m1.1, data = m1.data, iter = 1000, chains = 2, cores = 2)
print(fit.m1.1, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: bb69f84792e053a2e0234193f5617c02.
## 2 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=1000.
## 
##           mean se_mean   sd      10%      50%      90% n_eff Rhat
## mu      154.64    0.02 0.41   154.08   154.63   155.18   612    1
## sigma     7.77    0.01 0.29     7.42     7.75     8.14   875    1
## lp__  -1223.11    0.04 0.94 -1224.36 -1222.83 -1222.23   467    1
## 
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:45:12 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

2.1.2 Notation 2 (vectorized with Jags-like notation)

using the sampling statement

Difference with explicitly incrementing the log probability function:

“The sampling statement drops all the terms in the log probability function that are constant, whereas the explicit call to normal_lpdf adds all of the terms in the definition of the log normal probability function, including all of the constant normalizing terms. Therefore, the explicit increment form can be used to recreate the exact log probability values for the model. Otherwise, the sampling statement form will be faster if any of the input expressions involve only constants, data variables, and transformed data variables.”

https://mc-stan.org/docs/2_18/reference-manual/sampling-statements-section.html

Bounding sigma in the parameter block:

data {                                             // observed variables 
  int<lower=1> N;                                  // number of observations
  vector[N] height;                                // response variables
}
parameters {                                       // unobserved variables
  real mu;                                      
  real<lower=0,upper=50> sigma;
}
model {
  height ~ normal(mu,sigma);                       // likelihood
  mu ~ normal(178, 20);                            // prior of the mean
}
fit.m1.2.1 <- sampling(m1.2.1, data = m1.data, iter = 1000, chains = 2, cores = 2)
print(fit.m1.2.1, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: 362449507f9771e1464afdb2ddaff73e.
## 2 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=1000.
## 
##          mean se_mean   sd     10%     50%     90% n_eff Rhat
## mu     154.62    0.01 0.41  154.12  154.62  155.17   892    1
## sigma    7.77    0.01 0.29    7.43    7.77    8.17   922    1
## lp__  -895.73    0.04 0.91 -897.01 -895.42 -894.86   542    1
## 
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:46:01 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

Specifying the prior of sigma in the model block (\(\sigma \sim \mathcal{U}(0,50)\)):

data {                                             // observed variables 
  int<lower=1> N;                                  // number of observations
  vector[N] height;                                // response variables
}
parameters {                                       // unobserved variables
  real mu;                                      
  real sigma;
}
model {
  height ~ normal(mu,sigma);                       // likelihood
  mu ~ normal(178, 20);                            // prior of the mean
  sigma ~ uniform(0,50);                           // prior of sigma          
}
fit.m1.2.2 <- sampling(m1.2.2, data = m1.data, iter = 1000, chains = 2, cores = 2)
print(fit.m1.2.2, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: 4ad4544d00fe0bd73bdaf4a55e08b243.
## 2 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=1000.
## 
##          mean se_mean   sd     10%     50%     90% n_eff Rhat
## mu     154.62    0.02 0.41  154.10  154.61  155.15   548    1
## sigma    7.78    0.01 0.30    7.40    7.77    8.17   551    1
## lp__  -897.63    0.05 1.01 -898.94 -897.30 -896.73   422    1
## 
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:46:47 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

2.1.3 Notation 3 (unvectorized)

data {                                             // observed variables 
  int<lower=1> N;                                  // number of observations
  vector[N] height;                                // response variables
}
parameters {                                       // unobserved variables
  real mu;                                      
  real<lower=0,upper=50> sigma;
}
model {
  for (n in 1:N)
  height[n] ~ normal(mu,sigma);                    // likelihood
  mu ~ normal(178, 20);                            // prior of the mean
}
fit.m1.3 <- sampling(m1.3, data = m1.data, iter = 1000, chains = 2, cores = 2)
print(fit.m1.3, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: 2c0d3bdaed726863b2342ed270e93080.
## 2 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=1000.
## 
##          mean se_mean   sd     10%     50%     90% n_eff Rhat
## mu     154.60    0.02 0.42  154.06  154.60  155.14   721    1
## sigma    7.77    0.01 0.30    7.40    7.76    8.18   776    1
## lp__  -895.76    0.04 0.94 -897.04 -895.48 -894.87   456    1
## 
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:47:37 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

2.1.3.1 Comparing model runtime

Vectorized with target += notation:

get_elapsed_time(fit.m1.1)
##           warmup   sample
## chain:1 0.013483 0.011086
## chain:2 0.018512 0.020148

Vectorized with Jags-like notation:

# not specifying the prior of sigma
get_elapsed_time(fit.m1.2.1)
##           warmup   sample
## chain:1 0.020005 0.021851
## chain:2 0.018100 0.012708
# Specifying the prior of sigma
get_elapsed_time(fit.m1.2.2)
##           warmup   sample
## chain:1 0.062646 0.015833
## chain:2 0.021128 0.010710

Unvectorized:

get_elapsed_time(fit.m1.3)
##           warmup   sample
## chain:1 0.074360 0.090645
## chain:2 0.099856 0.095281

2.2 With low variance

\[ h_{i} \sim \mathcal{N}(\mu \, , \, \sigma^2)\] \[\mu \sim \mathcal{N}(178,0.1)\]

Data in a list:

m2.data <- list(N = nrow(howell2), height = howell2$height)
data {                                             // observed variables 
  int<lower=1> N;                                  // number of observations
  vector[N] height;                                // response variables
}
parameters {                                       // unobserved variables
  real mu;                                      
  real<lower=0,upper=50> sigma;
}
model {
  target += normal_lpdf(height | mu, sigma);       // likelihood
  target += normal_lpdf(mu | 178, 0.1);             // prior of the mean
}
fit.m2 <- sampling(m2 , data = m2.data, iter = 1000, chains = 2, cores = 2)
print(fit.m2, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: 732cdcf02bdc315da3f0b397f4c17596.
## 2 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=1000.
## 
##           mean se_mean   sd      10%      50%      90% n_eff Rhat
## mu      177.86    0.00 0.09   177.74   177.86   177.99   779    1
## sigma    24.56    0.03 0.94    23.37    24.57    25.74   827    1
## lp__  -1623.61    0.05 0.93 -1624.84 -1623.32 -1622.79   365    1
## 
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:48:28 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

3 With one continious predictor

Model equation:

\[\begin{align*} h_{i} \sim & \, \mathcal{N}(\mu_{i} \, , \, \sigma^2) \\[4pt] \mu_{i} = & \, \beta_{1} + \beta_{2} w_{i}\\[4pt] \beta_{1} \sim & \, \mathcal{N}(178,100)\\[4pt] \beta_{2} \sim & \, \mathcal{N}(0,10)\\[4pt] \sigma \sim & \, \mathcal{U}(0,5)\\[4pt] \end{align*}\]

Centering the predictor:

howell2$weight.c <- howell2$weight - mean(howell2$weight)

Data in a list:

m3.data <- list(N = nrow(howell2), height = howell2$height, weight = howell2$weight.c)
data {                                                     // observed variables 
  int<lower=1> N;                                          // number of observations
  vector[N] height;                                        // response variables
  vector[N] weight;                                        // predictor (previously centered)
}
parameters {                                               // unobserved variables
  real beta[2];                                            // parameters: intercept and slopes
  real sigma;
}
model {
  height ~ normal(beta[1] + beta[2]*weight ,sigma);        // likelihood
  beta[1] ~ normal(178, 100);                              // prior of the intercept
  beta[2] ~ normal(0, 10);                                 // prior of the slope
  sigma ~ uniform(0,50);
}
fit.m3.1 <- sampling(m3.1, data = m3.data, iter = 1000, chains = 4, cores = 4)
print(fit.m3.1, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: dca6bb84ed30c87a7015873cc74526bf.
## 4 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=2000.
## 
##            mean se_mean   sd     10%     50%     90% n_eff Rhat
## beta[1]  154.60    0.01 0.29  154.23  154.59  154.96  1841    1
## beta[2]    0.90    0.00 0.04    0.85    0.90    0.96   372    1
## sigma      5.11    0.01 0.20    4.87    5.10    5.37  1438    1
## lp__    -749.13    0.05 1.33 -750.77 -748.77 -747.87   600    1
## 
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:49:21 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

Another possibility for \(\beta_{2}\) prior: \[\beta_{2} \sim \, \text{LogNormal}(0,10)\].

data {                                                     // observed variables 
  int<lower=1> N;                                          // number of observations
  vector[N] height;                                        // response variables
  vector[N] weight;                                        // predictor (previously centered)
}
parameters {                                               // unobserved variables
  real beta[2];                                            // parameters: intercept and slopes
  real sigma;
}
model {
  height ~ normal(beta[1] + beta[2]*weight ,sigma);        // likelihood
  beta[1] ~ normal(178, 100);                              // prior of the intercept
  beta[2] ~ lognormal(0, 1);                               // prior of the slope
  sigma ~ uniform(0,50);
}
fit.m3.2 <- sampling(m3.2, data = m3.data, iter = 1000, chains = 4, cores = 4)
## Warning: There were 187 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
## http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
## Warning: There were 2 chains where the estimated Bayesian Fraction of Missing Information was low. See
## http://mc-stan.org/misc/warnings.html#bfmi-low
## Warning: Examine the pairs() plot to diagnose sampling problems
print(fit.m3.2, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: 76e2c520e9a282812337c63011f1f5e3.
## 4 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=2000.
## 
##            mean se_mean     sd     10%     50%     90% n_eff Rhat
## beta[1]  153.38    1.49  10.27  153.99  154.59  155.09    48 1.06
## beta[2]    0.90    0.00   0.11    0.82    0.90    0.98   978 1.00
## sigma      9.82    3.22   9.68    4.94    5.21   15.10     9 1.36
## lp__    -844.87   66.89 193.61 -975.39 -749.26 -747.81     8 1.36
## 
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:50:06 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

vectorized with target +=

data {                                                     // observed variables 
  int<lower=1> N;                                          // number of observations
  vector[N] height;                                        // response variables
  vector[N] weight;                                        // predictor (previously centered)
}
parameters {                                               // unobserved variables
  real beta[2];                                            // parameters: intercept and slopes
  real sigma;
}
model {
  target += normal_lpdf(height | beta[1] + beta[2]*weight, sigma);       // likelihood
  target += normal_lpdf(beta[1] | 178, 20);                              // prior of the intercept
  target += lognormal_lpdf(beta[2] |0, 1);                                // prior of the slope
  target += uniform_lpdf(sigma|0,50);
}
fit.m3.3 <- sampling(m3.3, data = m3.data, iter = 1000, chains = 4, cores = 4)
## Warning: There were 123 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
## http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
## Warning: There were 1 chains where the estimated Bayesian Fraction of Missing Information was low. See
## http://mc-stan.org/misc/warnings.html#bfmi-low
## Warning: Examine the pairs() plot to diagnose sampling problems
print(fit.m3.3, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: c0bd04f9d695ff94776043871bd4d625.
## 4 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=2000.
## 
##             mean se_mean     sd      10%      50%      90% n_eff  Rhat
## beta[1]   154.60    0.02   0.67   154.00   154.60   155.17  1102  1.00
## beta[2]     0.89    0.01   0.09     0.79     0.89     0.97   164  1.06
## sigma       9.47    5.34   7.62     4.90     5.19    22.09     2  8.04
## lp__    -1170.98  109.09 154.85 -1433.29 -1082.10 -1080.69     2 13.21
## 
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:50:52 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

3.1 What do our priors imply?

Let’s simulate the prior predictive distribution (simulate heights from the model, using only the priors).

See Chapter 4 of Statistical Rethinking.

set.seed(2971)
N <- 100 # 100 lines
a <- rnorm( N , 178 , 20 )
b1 <- rnorm( N , 0 , 10 )
b2 <- rlnorm( N, 0 , 1 )
dens(rnorm(  1e4, 0 , 10 ),main="b ~ dnorm(0,10)")

dens(rlnorm( 1e4 , 0 , 1 ),main="log(b) ~ dnorm(0,1)")

p <- plot( NULL , xlim=range(howell2$weight) , ylim=c(-100,400) ,xlab="weight" , ylab="height")
abline(h=0,lty=2)
abline( h=272 , lty=1 , lwd=0.5 )
mtext( "b ~ dnorm(0,10)" )
xbar <- mean(howell2$weight)
for ( i in 1:N ) curve( a[i] + b1[i]*(x - xbar) ,
from=min(howell2$weight) , to=max(howell2$weight) , add=TRUE ,
col=col.alpha("black",0.2) )

McElreath: “For reference, I’ve added a dashed line at zero—no one is shorter than zero—and the “Wadlow” line at 272cm for the world’s tallest person. The pattern doesn’t look like any human population at all. It essentially says that the relationship between weight and height could be absurdly positive or negative. Before we’ve even seen the data, this is a bad model."

plot( NULL , xlim=range(howell2$weight) , ylim=c(-100,400) ,xlab="weight" , ylab="height")
abline(h=0,lty=2)
abline( h=272 , lty=1 , lwd=0.5 )
mtext( "log(b) ~ dnorm(0,1)" )
xbar <- mean(howell2$weight)
for ( i in 1:N ) curve( a[i] + b2[i]*(x - xbar) ,
from=min(howell2$weight) , to=max(howell2$weight) , add=TRUE ,
col=col.alpha("black",0.2) )

McElreath: “This is much more sensible. There is still a rare impossible relationship. But nearly all lines in the joint prior for \(\beta_{1}\) and \(\beta_{2}\) are now within human reason. We’re fussing about this prior, even though as you’ll see in the next section there is so much data in this example that the priors end up not mattering. We fuss for two reasons. First, there are many analyses in which no amount of data makes the prior irrelevant. In such cases, non-Bayesian procedures are no better off. They also depend upon structural features of the model. Paying careful attention to those features is essential. Second, thinking about the priors helps us develop better models, maybe even eventually going beyond geocentrism.”

Help to choose your prior with Stan: https://github.com/stan-dev/stan/wiki/Prior-Choice-Recommendations

3.2 Convergence diagnostics

https://github.com/stan-dev/stan/wiki/Stan-Best-Practices

3.2.1 Rhat < 1.1

“The best generic diagnostic for the accuracy of a Markov chain Monte Carlo algorithm is the split R-hat statistic which quantifies the consistency of an ensemble of Markov chains. The idea is to run multiple Markov chains, 4 is an okay default but running more makes the diagnostic more sensitive, and ensure that they’re all exploring the same regions of parameter space. If even one chain is inconsistent with the others then all of the chains are suspect! In practice we have found that requiring Rhat < 1.1 is a good default requirement for each parameter.”

rhats <- rhat(fit.m3.2)
print(rhats)
##   beta[1]   beta[2]     sigma      lp__ 
## 1.0578017 0.9999526 1.3590584 1.3611599
mcmc_rhat(rhats)  + yaxis_text(hjust = 1)

3.2.2 Effective sample size

“Another potential problem to keep in mind is Markov chains that explore very slowly. Slow chains lead to low numbers of effective samples, and if there are too few effective samples then we can’t accurate estimate the number of effective samples at all. A good check for such issues is the number of effective samples per iteration – if N_eff / N < 0.001 then you should be suspect of the effective sample size calculation.”

ratios <- neff_ratio(fit.m3.2)
print(ratios)
##     beta[1]     beta[2]       sigma        lp__ 
## 0.023845504 0.488859828 0.004527409 0.004189118
mcmc_neff(ratios, size = 2)

3.2.3 MCMC diagnostics using the bayesplot package

We need:

# The posterior draws
posterior <- as.array(fit.m3.2)

# The  log of the posterior density for each draw 
lp <- log_posterior(fit.m3.2)
head(lp)
##   Iteration     Value Chain
## 1         1 -748.2942     1
## 2         2 -748.4240     1
## 3         3 -747.9331     1
## 4         4 -747.7890     1
## 5         5 -748.0312     1
## 6         6 -751.1590     1
# Some NUTS-specific diagnostic values 
np <- nuts_params(fit.m3.2)
head(np)
##   Iteration     Parameter     Value Chain
## 1         1 accept_stat__ 0.5872795     1
## 2         2 accept_stat__ 0.9966800     1
## 3         3 accept_stat__ 0.9712667     1
## 4         4 accept_stat__ 0.9852274     1
## 5         5 accept_stat__ 0.9135522     1
## 6         6 accept_stat__ 0.7162124     1

3.2.3.1 Divergent transitions

https://mc-stan.org/bayesplot/articles/visual-mcmc-diagnostics.html

mcmc_parcoord(posterior, np = np,pars=c("beta[1]","beta[2]","sigma"))

Pairs plots:

mcmc_pairs(posterior, np = np, pars = c("beta[1]","beta[2]","sigma"),  off_diag_args = list(size = 0.75))

Trace plots:

mcmc_trace(posterior, pars = "beta[2]", np = np) +  xlab("Post-warmup iteration")

3.3 Posterior predictive distributions

https://mc-stan.org/docs/2_18/stan-users-guide/prediction-forecasting-and-backcasting.html

m4.data <- list(N = nrow(howell2), height = howell2$height, weight = howell2$weight.c)
data {                                                     // observed variables 
  int<lower=1> N;                                          // number of observations
  vector[N] height;                                        // response variables
  vector[N] weight;                                        // predictor (previously centered)
}

parameters {                                               // unobserved variables
  real beta[2];                                            // parameters: intercept and slope
  real sigma;
}

model {
  height ~ normal(beta[1] + beta[2]*weight ,sigma);        // likelihood
  beta[1] ~ normal(178, 100);                              // prior of the intercept
  beta[2] ~ lognormal(0, 1);                                  // prior of the slope
  sigma ~ uniform(0,50);
}

generated quantities {
  vector[N] y_rep;

  for(i in 1:N)
  y_rep[i] = normal_rng(beta[1] + beta[2]*weight[i], sigma);
}
fit.m4 <- sampling(m4, data = m4.data, iter = 1000, chains = 4, cores = 4)
y_rep <- as.matrix(fit.m4, pars = "y_rep")
dim(y_rep)
## [1] 2000  352
ppc_dens_overlay(howell2$height, y_rep)

plot_grid(ppc_stat(y = howell2$height, yrep = y_rep, stat = "mean"),
          ppc_scatter_avg(y = howell2$height, yrep = y_rep))

3.4 Visualizing the variance

See section 4.4 of Statistical rethinking

Model with non-centered weight:

m3.1.bis.data <- list(N = nrow(howell2), height = howell2$height, weight = howell2$weight)
data {                                                     // observed variables 
  int<lower=1> N;                                          // number of observations
  vector[N] height;                                        // response variables
  vector[N] weight;                                        // predictor (previously centered)
}
parameters {                                               // unobserved variables
  real beta[2];                                            // parameters: intercept and slopes
  real sigma;
}
model {
  height ~ normal(beta[1] + beta[2]*weight ,sigma);        // likelihood
  beta[1] ~ normal(178, 100);                              // prior of the intercept
  beta[2] ~ normal(0, 10);                                 // prior of the slope
  sigma ~ uniform(0,50);
}
fit.m3.1.bis <- sampling(m3.1.bis, data = m3.1.bis.data, iter = 1000, chains = 4, cores = 4)
print(fit.m3.1.bis, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: dca6bb84ed30c87a7015873cc74526bf.
## 4 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=2000.
## 
##            mean se_mean   sd     10%     50%     90% n_eff Rhat
## beta[1]  113.95    0.09 2.08  111.29  113.92  116.54   517    1
## beta[2]    0.90    0.00 0.05    0.85    0.90    0.96   527    1
## sigma      5.10    0.01 0.20    4.86    5.10    5.35   844    1
## lp__    -749.35    0.05 1.31 -751.13 -749.01 -748.05   578    1
## 
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:52:33 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

3.4.1 Prediction interval for average height

Figure 4.7. of Statistical rethinking

post <- as.data.frame(fit.m3.1.bis)
f_mu <- function(x) post$`beta[1]` + post$`beta[2]` * x
weight_new <- seq(25, 70)
mu <- 
  sapply(weight_new, f_mu) %>%
  tibble::as_tibble() %>%
  rename_all(function(x) weight_new) %>%
  mutate(Iter = row_number()) %>%
  gather(weight, height, -Iter) %>%
  group_by(weight) %>%
  mutate(hpdi_l = HDInterval::hdi(height, credMass = 0.89)[1],
         hpdi_h = HDInterval::hdi(height, credMass = 0.89)[2]) %>%
  mutate(mu = mean(height)) %>%
  ungroup() %>%
  mutate(weight = as.integer(weight))
p <- ggplot() 
p1 <- p +
  geom_point(data = mu %>% filter(Iter < 101),
             aes(weight, height), alpha = .05, color = 'dodgerblue') +
  labs(subtitle="Density at each weight")
p2 <- p +
  geom_point(data = howell2,
             aes(weight, height), shape = 1, color = 'dodgerblue') +
  geom_ribbon(data = mu,
              aes(x = weight, ymin = hpdi_l, ymax = hpdi_h),
              alpha = .2) +
  geom_abline(data = post,
              aes(intercept = mean(`beta[1]`), slope = mean(`beta[2]`))) +
  labs(subtitle="HPDI Interval = 0.89")
grid.arrange(p1, p2, nrow = 1)  

“Left: The first 100 values in the distribution of \(\beta_{1}\) (intercept) at each weight value. Right: The !Kung height data again, now with 89% HPDI of the mean indicated by the shaded region. Compare this region to the distributions of blue points on the left.”

3.4.2 Prediction interval for actual heights

Figure 4.8. of Statistical rethinking

Incorporating the uncertainty related to the standard deviation \(\sigma^{2}\).

sim_ht <- 
  sapply(weight_new,
         function(x)
           rnorm(NROW(post),
                 post$`beta[1]` + post$`beta[2]` * x,
                 post$sigma)) %>%
  as_tibble() %>%
  rename_all(function(x) weight_new) %>%
  mutate(Iter = row_number()) %>%
  gather(weight, height, -Iter) %>%
  group_by(weight) %>%
  mutate(pi_l = rethinking::PI(height, prob = 0.89)[1],
         pi_h = rethinking::PI(height, prob = 0.89)[2]) %>%
  ungroup() %>%
  mutate(weight = as.integer(weight))
p2 + geom_ribbon(data = sim_ht,
                 mapping = aes(x=weight, ymin=pi_l, ymax=pi_h), alpha = .1) +
  labs(subtitle = 'Prediction Intervals = 0.89')

Figure 4.8. 89% prediction interval for height, as a function of weight. The solid line is the MAP estimate of the mean height at each weight. The two shaded regions show different 89% plausible regions. The narrow shaded interval around the line is the distribution of \(\beta_{1}\) (intercept) . The wider shaded region represents the region within which the model expects to find 89% of actual heights in the population, at each weight.”"

4 With one categorical predictor

Individual sex is a categorical variable with value 0 whenever the person is a female and a value 1 whenever the person is a male.

ggplot(howell2, aes(x=as.factor(male), y=height, color=as.factor(male))) +
  geom_boxplot()   + scale_x_discrete(labels=c("0" = "Female", "1" = "Male")) + 
  theme(axis.title.x=element_blank(),legend.position = "none")

There are two ways to include binary categorical variables, either using indicator variable or index variable. More details in section 5.3.1 of Statistical rethinking (McElreath’s book).

4.1 Indicator variable (or dummy variable)

Using indicator variables is ok with binary variables, but shouldn’t be used for variables with many categories (see 5.3.2 section of Statistical rethiking).

Model equation (like in McElreath’s book):

\[\begin{align*} h_{i} &\sim \mathcal{N}(\mu_{i} \, , \, \sigma^2) \\[4pt] \mu_{i} &= \alpha + \beta_{m} m_{i}\\[4pt] \alpha &\sim \mathcal{N}(178,20)\\[4pt] \beta_{m} &\sim \mathcal{N}(0,10)\\[4pt] \sigma &\sim \mathcal{U}(0,50)\\[4pt] \end{align*}\]

In R, we can convert our categorical variable in a dummy variable like this:

X <- model.matrix(~male, howell2)
m5.1.data <- list(N = nrow(howell2), height = howell2$height, X = X, n = length(unique(howell2$male)))
str(m5.1.data)
## List of 4
##  $ N     : int 352
##  $ height: num [1:352] 152 140 137 157 145 ...
##  $ X     : num [1:352, 1:2] 1 1 1 1 1 1 1 1 1 1 ...
##   ..- attr(*, "dimnames")=List of 2
##   .. ..$ : chr [1:352] "1" "2" "3" "4" ...
##   .. ..$ : chr [1:2] "(Intercept)" "male"
##   ..- attr(*, "assign")= int [1:2] 0 1
##  $ n     : int 2
data {                                                     
  int<lower=0> N;                                          // number of observations
  int<lower=0> n;                                          // number of levels of the categorical variable
  vector[N] height;                                        // response variables
  matrix[N,n] X;                                           // categorical variable
}

parameters {                                          
  vector[n] beta;                                         // intercepts of the categorical variable
  real sigma;
}

transformed parameters {
vector[N] mu;                                             
mu = X*beta;  
}

model {
  height ~ normal(mu ,sigma);                              // likelihood
  sigma ~ uniform(0,50);
  beta[1] ~ normal(178,20);
  beta[2] ~ normal(0,10);
}

generated quantities {
  vector[N] y_rep;
 for(i in 1:N)
 y_rep[i] = normal_rng(mu[i], sigma);
}
fit.m5_1_1 <- sampling(m5.1.1, data = m5.1.data, iter = 1000, chains = 4, cores = 4)
print(fit.m5_1_1, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: a89cc41ee4ed1022b476f4484ab5720d.
## 4 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=2000.
## 
##               mean se_mean   sd     10%     50%     90% n_eff Rhat
## beta[1]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## beta[2]      10.82    0.03 0.53   10.15   10.82   11.51   392 1.01
## sigma         5.55    0.00 0.21    5.27    5.54    5.82  2099 1.00
## mu[1]       160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[2]       149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[3]       149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[4]       160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[5]       149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[6]       160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[7]       149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[8]       160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[9]       149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[10]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[11]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[12]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[13]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[14]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[15]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[16]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[17]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[18]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[19]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[20]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[21]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[22]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[23]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[24]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[25]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[26]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[27]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[28]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[29]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[30]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[31]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[32]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[33]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[34]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[35]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[36]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[37]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[38]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[39]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[40]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[41]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[42]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[43]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[44]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[45]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[46]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[47]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[48]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[49]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[50]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[51]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[52]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[53]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[54]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[55]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[56]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[57]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[58]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[59]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[60]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[61]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[62]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[63]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[64]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[65]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[66]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[67]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[68]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[69]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[70]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[71]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[72]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[73]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[74]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[75]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[76]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[77]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[78]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[79]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[80]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[81]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[82]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[83]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[84]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[85]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[86]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[87]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[88]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[89]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[90]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[91]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[92]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[93]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[94]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[95]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[96]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[97]      149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[98]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[99]      160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[100]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[101]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[102]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[103]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[104]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[105]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[106]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[107]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[108]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[109]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[110]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[111]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[112]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[113]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[114]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[115]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[116]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[117]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[118]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[119]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[120]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[121]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[122]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[123]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[124]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[125]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[126]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[127]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[128]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[129]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[130]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[131]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[132]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[133]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[134]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[135]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[136]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[137]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[138]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[139]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[140]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[141]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[142]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[143]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[144]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[145]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[146]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[147]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[148]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[149]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[150]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[151]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[152]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[153]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[154]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[155]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[156]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[157]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[158]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[159]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[160]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[161]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[162]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[163]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[164]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[165]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[166]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[167]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[168]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[169]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[170]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[171]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[172]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[173]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[174]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[175]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[176]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[177]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[178]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[179]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[180]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[181]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[182]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[183]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[184]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[185]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[186]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[187]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[188]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[189]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[190]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[191]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[192]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[193]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[194]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[195]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[196]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[197]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[198]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[199]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[200]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[201]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[202]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[203]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[204]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[205]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[206]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[207]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[208]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[209]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[210]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[211]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[212]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[213]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[214]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[215]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[216]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[217]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[218]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[219]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[220]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[221]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[222]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[223]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[224]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[225]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[226]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[227]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[228]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[229]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[230]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[231]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[232]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[233]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[234]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[235]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[236]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[237]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[238]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[239]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[240]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[241]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[242]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[243]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[244]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[245]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[246]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[247]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[248]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[249]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[250]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[251]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[252]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[253]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[254]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[255]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[256]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[257]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[258]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[259]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[260]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[261]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[262]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[263]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[264]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[265]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[266]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[267]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[268]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[269]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[270]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[271]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[272]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[273]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[274]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[275]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[276]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[277]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[278]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[279]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[280]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[281]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[282]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[283]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[284]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[285]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[286]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[287]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[288]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[289]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[290]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[291]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[292]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[293]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[294]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[295]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[296]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[297]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[298]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[299]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[300]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[301]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[302]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[303]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[304]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[305]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[306]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[307]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[308]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[309]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[310]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[311]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[312]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[313]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[314]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[315]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[316]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[317]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[318]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[319]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[320]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[321]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[322]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[323]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[324]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[325]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[326]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[327]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[328]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[329]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[330]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[331]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[332]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[333]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[334]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[335]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[336]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[337]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[338]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[339]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[340]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[341]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[342]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[343]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[344]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[345]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[346]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[347]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[348]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[349]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[350]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## mu[351]     149.53    0.02 0.38  149.05  149.53  150.02   594 1.01
## mu[352]     160.35    0.01 0.39  159.86  160.35  160.86   777 1.00
## y_rep[1]    160.43    0.13 5.56  153.29  160.31  167.57  1777 1.00
## y_rep[2]    149.55    0.13 5.57  142.44  149.54  156.74  1712 1.00
## y_rep[3]    149.53    0.12 5.52  142.47  149.52  156.76  2041 1.00
## y_rep[4]    160.19    0.13 5.63  153.02  160.31  167.40  1944 1.00
## y_rep[5]    149.29    0.12 5.48  142.35  149.03  156.36  1932 1.00
## y_rep[6]    160.36    0.13 5.66  153.26  160.19  167.56  1905 1.00
## y_rep[7]    149.50    0.12 5.47  142.61  149.52  156.20  1988 1.00
## y_rep[8]    160.38    0.12 5.50  153.32  160.63  167.19  2064 1.00
## y_rep[9]    149.44    0.13 5.67  142.25  149.29  156.69  1854 1.00
## y_rep[10]   160.43    0.13 5.75  153.26  160.33  168.07  1933 1.00
## y_rep[11]   149.69    0.12 5.55  142.55  149.82  156.65  2163 1.00
## y_rep[12]   160.25    0.12 5.51  153.24  160.19  167.27  2069 1.00
## y_rep[13]   149.45    0.12 5.65  142.31  149.25  156.78  2086 1.00
## y_rep[14]   149.55    0.12 5.49  142.54  149.50  156.52  1989 1.00
## y_rep[15]   149.37    0.13 5.70  142.21  149.36  156.89  2056 1.00
## y_rep[16]   160.36    0.13 5.50  153.15  160.43  167.48  1772 1.00
## y_rep[17]   160.42    0.13 5.52  153.22  160.43  167.63  1690 1.00
## y_rep[18]   149.33    0.13 5.63  142.09  149.47  156.41  1935 1.00
## y_rep[19]   160.13    0.13 5.66  152.96  160.27  167.39  2028 1.00
## y_rep[20]   149.50    0.12 5.49  142.40  149.66  156.47  2013 1.00
## y_rep[21]   160.31    0.13 5.44  153.19  160.44  167.30  1780 1.00
## y_rep[22]   149.26    0.12 5.57  142.06  149.41  156.31  2002 1.00
## y_rep[23]   149.66    0.13 5.61  142.46  149.71  156.92  1962 1.00
## y_rep[24]   160.23    0.12 5.55  153.44  160.16  167.23  2031 1.00
## y_rep[25]   160.47    0.13 5.57  153.24  160.63  167.76  1980 1.00
## y_rep[26]   149.51    0.13 5.62  142.39  149.54  156.64  1899 1.00
## y_rep[27]   149.70    0.14 5.65  142.53  149.71  156.92  1719 1.00
## y_rep[28]   149.41    0.12 5.55  142.34  149.45  156.43  1984 1.00
## y_rep[29]   149.80    0.14 5.58  142.46  149.90  157.04  1633 1.00
## y_rep[30]   149.70    0.13 5.44  142.78  149.73  156.62  1800 1.00
## y_rep[31]   149.54    0.13 5.49  142.53  149.52  156.63  1709 1.00
## y_rep[32]   160.27    0.13 5.64  153.03  160.16  167.65  1758 1.00
## y_rep[33]   160.35    0.13 5.73  153.06  160.19  167.81  1930 1.00
## y_rep[34]   160.02    0.13 5.50  152.80  160.11  166.93  1934 1.00
## y_rep[35]   160.24    0.13 5.42  153.35  160.33  167.01  1849 1.00
## y_rep[36]   149.55    0.12 5.49  142.44  149.56  156.72  2217 1.00
## y_rep[37]   149.57    0.14 5.52  142.49  149.48  156.61  1633 1.00
## y_rep[38]   160.46    0.12 5.47  153.25  160.48  167.28  2041 1.00
## y_rep[39]   149.65    0.12 5.57  142.37  149.66  156.85  2042 1.00
## y_rep[40]   160.11    0.13 5.65  152.87  160.19  167.26  2016 1.00
## y_rep[41]   160.33    0.12 5.55  153.04  160.40  167.53  2044 1.00
## y_rep[42]   160.49    0.12 5.49  153.46  160.56  167.40  2114 1.00
## y_rep[43]   149.61    0.12 5.59  142.67  149.52  156.86  2076 1.00
## y_rep[44]   149.52    0.13 5.64  142.13  149.64  156.56  2012 1.00
## y_rep[45]   149.60    0.13 5.41  142.63  149.54  156.81  1844 1.00
## y_rep[46]   149.54    0.13 5.58  142.49  149.50  156.80  1792 1.00
## y_rep[47]   160.54    0.14 5.54  153.54  160.53  167.81  1625 1.00
## y_rep[48]   149.42    0.12 5.58  142.04  149.60  156.59  2043 1.00
## y_rep[49]   160.32    0.13 5.55  153.51  160.17  167.51  1867 1.00
## y_rep[50]   149.69    0.12 5.53  142.41  149.70  156.79  2010 1.00
## y_rep[51]   149.43    0.13 5.73  141.94  149.44  156.60  2003 1.00
## y_rep[52]   149.29    0.13 5.72  141.63  149.28  156.49  2009 1.00
## y_rep[53]   160.31    0.12 5.45  153.27  160.33  167.33  2007 1.00
## y_rep[54]   160.35    0.13 5.62  153.27  160.38  167.57  1941 1.00
## y_rep[55]   149.60    0.13 5.52  142.65  149.50  157.01  1877 1.00
## y_rep[56]   160.29    0.12 5.45  153.17  160.26  167.04  1903 1.00
## y_rep[57]   149.47    0.13 5.53  142.35  149.46  156.58  1798 1.00
## y_rep[58]   160.25    0.13 5.62  152.84  160.35  167.23  1877 1.00
## y_rep[59]   149.68    0.12 5.64  142.44  149.75  156.78  2171 1.00
## y_rep[60]   160.30    0.12 5.33  153.58  160.26  167.00  2057 1.00
## y_rep[61]   149.49    0.12 5.69  142.19  149.52  156.81  2087 1.00
## y_rep[62]   149.74    0.13 5.65  142.40  149.74  156.82  1814 1.00
## y_rep[63]   160.30    0.14 5.63  153.43  160.19  167.51  1723 1.00
## y_rep[64]   160.42    0.13 5.65  152.99  160.54  167.66  1989 1.00
## y_rep[65]   149.55    0.12 5.50  142.41  149.44  156.81  1951 1.00
## y_rep[66]   160.57    0.12 5.50  153.55  160.68  167.44  1991 1.00
## y_rep[67]   149.66    0.12 5.45  142.81  149.51  156.55  1964 1.00
## y_rep[68]   149.72    0.13 5.74  142.31  149.79  157.17  1888 1.00
## y_rep[69]   160.39    0.13 5.60  153.08  160.50  167.62  1872 1.00
## y_rep[70]   149.59    0.13 5.72  142.09  149.81  156.50  1960 1.00
## y_rep[71]   160.26    0.13 5.52  153.09  160.25  167.36  1944 1.00
## y_rep[72]   160.35    0.13 5.63  153.22  160.29  167.54  1827 1.00
## y_rep[73]   160.49    0.13 5.48  153.47  160.42  167.51  1906 1.00
## y_rep[74]   149.59    0.14 5.77  142.27  149.52  156.97  1669 1.00
## y_rep[75]   149.50    0.13 5.49  142.41  149.42  156.56  1904 1.00
## y_rep[76]   149.45    0.12 5.28  142.72  149.47  156.18  1924 1.00
## y_rep[77]   149.68    0.13 5.58  142.70  149.71  156.88  1811 1.00
## y_rep[78]   160.36    0.13 5.61  153.25  160.41  167.46  1866 1.00
## y_rep[79]   149.86    0.13 5.54  142.55  149.89  156.95  1884 1.00
## y_rep[80]   149.54    0.12 5.49  142.48  149.57  156.65  1981 1.00
## y_rep[81]   160.43    0.12 5.41  153.61  160.52  167.20  2023 1.00
## y_rep[82]   160.23    0.13 5.56  153.04  160.31  167.35  1898 1.00
## y_rep[83]   149.56    0.13 5.59  142.71  149.60  156.51  1976 1.00
## y_rep[84]   160.48    0.13 5.54  153.37  160.52  167.46  1830 1.00
## y_rep[85]   149.45    0.12 5.51  142.35  149.44  156.51  1985 1.00
## y_rep[86]   160.32    0.13 5.45  153.18  160.36  167.18  1843 1.00
## y_rep[87]   149.54    0.13 5.68  142.41  149.54  156.78  2050 1.00
## y_rep[88]   160.42    0.12 5.38  153.51  160.37  167.63  2177 1.00
## y_rep[89]   149.50    0.12 5.39  142.61  149.57  156.49  1911 1.00
## y_rep[90]   160.26    0.13 5.57  153.11  160.19  167.54  1784 1.00
## y_rep[91]   160.10    0.13 5.62  152.88  160.10  167.38  1959 1.00
## y_rep[92]   160.52    0.13 5.56  153.31  160.46  167.78  1972 1.00
## y_rep[93]   149.37    0.12 5.43  142.59  149.35  156.30  2008 1.00
## y_rep[94]   149.83    0.13 5.58  142.88  149.80  157.03  1942 1.00
## y_rep[95]   160.48    0.13 5.52  153.16  160.38  167.63  1936 1.00
## y_rep[96]   160.24    0.14 5.63  152.83  160.29  167.51  1644 1.00
## y_rep[97]   149.39    0.12 5.46  142.46  149.31  156.46  1924 1.00
## y_rep[98]   160.45    0.13 5.45  153.46  160.36  167.25  1882 1.00
## y_rep[99]   160.32    0.13 5.55  153.25  160.28  167.65  1854 1.00
## y_rep[100]  149.62    0.13 5.62  142.55  149.64  156.85  1988 1.00
## y_rep[101]  160.42    0.12 5.55  153.30  160.41  167.40  2017 1.00
## y_rep[102]  149.49    0.13 5.56  142.07  149.58  156.56  1837 1.00
## y_rep[103]  149.43    0.12 5.45  142.38  149.43  156.55  2231 1.00
## y_rep[104]  160.36    0.13 5.46  153.28  160.31  167.39  1643 1.00
## y_rep[105]  160.43    0.12 5.57  153.58  160.29  167.55  1994 1.00
## y_rep[106]  149.29    0.12 5.56  142.19  149.38  156.52  2155 1.00
## y_rep[107]  149.38    0.12 5.60  141.98  149.51  156.26  2062 1.00
## y_rep[108]  149.52    0.12 5.56  142.54  149.50  156.80  2069 1.00
## y_rep[109]  160.26    0.13 5.49  153.56  160.13  167.46  1894 1.00
## y_rep[110]  149.46    0.12 5.56  142.21  149.40  156.54  1988 1.00
## y_rep[111]  160.44    0.13 5.65  152.97  160.50  167.76  2032 1.00
## y_rep[112]  149.75    0.12 5.52  142.73  149.58  156.83  1968 1.00
## y_rep[113]  149.45    0.13 5.63  142.08  149.55  156.56  1880 1.00
## y_rep[114]  160.42    0.13 5.66  153.13  160.20  167.68  1859 1.00
## y_rep[115]  160.14    0.12 5.49  152.98  160.20  167.05  2131 1.00
## y_rep[116]  149.48    0.13 5.75  142.29  149.39  156.86  1956 1.00
## y_rep[117]  160.27    0.13 5.57  153.30  160.31  167.30  1939 1.00
## y_rep[118]  160.30    0.12 5.44  153.55  160.24  167.26  1934 1.00
## y_rep[119]  149.36    0.12 5.48  142.41  149.33  156.25  1958 1.00
## y_rep[120]  149.43    0.13 5.65  142.35  149.34  156.70  1846 1.00
## y_rep[121]  160.26    0.13 5.59  153.00  160.37  167.33  1974 1.00
## y_rep[122]  149.48    0.13 5.71  142.33  149.54  156.63  1978 1.00
## y_rep[123]  160.15    0.13 5.68  152.81  160.23  167.39  1852 1.00
## y_rep[124]  149.51    0.12 5.46  142.51  149.46  156.57  2063 1.00
## y_rep[125]  149.52    0.13 5.44  142.75  149.57  156.34  1760 1.00
## y_rep[126]  149.36    0.13 5.62  142.20  149.27  156.48  2005 1.00
## y_rep[127]  149.53    0.13 5.73  142.30  149.59  156.97  1884 1.00
## y_rep[128]  160.26    0.13 5.58  152.86  160.29  167.49  1978 1.00
## y_rep[129]  149.58    0.13 5.34  142.28  149.65  156.29  1656 1.00
## y_rep[130]  149.57    0.12 5.46  142.57  149.59  156.62  1990 1.00
## y_rep[131]  160.34    0.13 5.65  153.25  160.32  167.62  1883 1.00
## y_rep[132]  160.24    0.12 5.52  153.32  160.05  167.38  2074 1.00
## y_rep[133]  149.64    0.13 5.53  142.54  149.63  156.96  1830 1.00
## y_rep[134]  160.53    0.12 5.60  153.20  160.52  167.48  2068 1.00
## y_rep[135]  149.36    0.12 5.62  141.94  149.43  156.45  2088 1.00
## y_rep[136]  160.41    0.13 5.76  153.26  160.42  167.77  1930 1.00
## y_rep[137]  160.28    0.13 5.71  153.15  160.10  167.60  1987 1.00
## y_rep[138]  160.32    0.13 5.38  153.38  160.27  167.07  1672 1.00
## y_rep[139]  149.52    0.13 5.67  142.16  149.48  156.88  1989 1.00
## y_rep[140]  160.43    0.13 5.59  153.33  160.35  167.56  1718 1.00
## y_rep[141]  149.44    0.12 5.60  142.03  149.40  156.56  2088 1.00
## y_rep[142]  149.74    0.12 5.52  142.49  149.82  156.76  1983 1.00
## y_rep[143]  149.39    0.12 5.52  142.23  149.30  156.42  2033 1.00
## y_rep[144]  149.55    0.12 5.70  142.63  149.45  156.60  2130 1.00
## y_rep[145]  149.52    0.12 5.59  142.20  149.59  156.60  2037 1.00
## y_rep[146]  160.56    0.12 5.59  153.36  160.52  167.48  2056 1.00
## y_rep[147]  149.63    0.13 5.67  142.24  149.64  156.88  1912 1.00
## y_rep[148]  149.38    0.13 5.54  142.20  149.30  156.44  1893 1.00
## y_rep[149]  160.49    0.13 5.64  153.39  160.38  167.54  1984 1.00
## y_rep[150]  149.43    0.12 5.68  142.22  149.20  156.86  2164 1.00
## y_rep[151]  160.69    0.13 5.64  153.47  160.67  168.08  1931 1.00
## y_rep[152]  149.63    0.12 5.54  142.46  149.63  156.92  2036 1.00
## y_rep[153]  160.32    0.13 5.62  153.19  160.51  167.54  1811 1.00
## y_rep[154]  149.43    0.12 5.41  142.71  149.33  156.40  2053 1.00
## y_rep[155]  160.40    0.13 5.71  152.87  160.41  167.52  2017 1.00
## y_rep[156]  149.52    0.12 5.56  142.50  149.63  156.79  2159 1.00
## y_rep[157]  149.33    0.12 5.56  142.42  149.41  156.47  2090 1.00
## y_rep[158]  160.40    0.13 5.53  153.41  160.41  167.48  1904 1.00
## y_rep[159]  160.49    0.12 5.58  153.28  160.62  167.63  2096 1.00
## y_rep[160]  149.37    0.13 5.52  142.35  149.24  156.49  1902 1.00
## y_rep[161]  160.12    0.12 5.54  152.81  160.10  167.29  2019 1.00
## y_rep[162]  149.40    0.13 5.62  142.41  149.35  156.57  1946 1.00
## y_rep[163]  160.43    0.12 5.50  153.57  160.36  167.36  2110 1.00
## y_rep[164]  160.36    0.13 5.65  153.04  160.51  167.60  1850 1.00
## y_rep[165]  160.43    0.12 5.50  153.45  160.39  167.39  1968 1.00
## y_rep[166]  149.38    0.13 5.67  142.21  149.35  156.63  1853 1.00
## y_rep[167]  160.40    0.13 5.79  152.83  160.52  167.76  2036 1.00
## y_rep[168]  149.64    0.13 5.56  142.65  149.67  156.72  1929 1.00
## y_rep[169]  149.49    0.13 5.62  142.25  149.33  156.79  1980 1.00
## y_rep[170]  149.60    0.13 5.52  142.48  149.73  156.72  1943 1.00
## y_rep[171]  149.66    0.12 5.49  142.59  149.62  156.67  1985 1.00
## y_rep[172]  160.34    0.14 5.60  153.09  160.52  167.18  1625 1.00
## y_rep[173]  149.20    0.12 5.53  141.92  149.16  156.22  1959 1.00
## y_rep[174]  160.41    0.13 5.63  153.28  160.42  167.68  1803 1.00
## y_rep[175]  149.43    0.12 5.56  142.25  149.33  156.70  2086 1.00
## y_rep[176]  160.32    0.12 5.67  152.95  160.32  167.54  2097 1.00
## y_rep[177]  149.44    0.12 5.62  142.26  149.46  156.63  2077 1.00
## y_rep[178]  149.49    0.12 5.57  142.75  149.57  156.45  2023 1.00
## y_rep[179]  160.44    0.13 5.69  153.16  160.47  167.78  2070 1.00
## y_rep[180]  149.38    0.13 5.68  142.01  149.29  156.73  1994 1.00
## y_rep[181]  160.47    0.13 5.66  153.18  160.49  167.61  2015 1.00
## y_rep[182]  149.53    0.13 5.70  142.29  149.30  157.14  1844 1.00
## y_rep[183]  149.72    0.13 5.56  142.75  149.77  156.83  1876 1.00
## y_rep[184]  149.66    0.12 5.48  142.73  149.50  156.68  1973 1.00
## y_rep[185]  160.28    0.13 5.54  153.14  160.37  167.24  1901 1.00
## y_rep[186]  160.38    0.13 5.48  153.27  160.38  167.51  1914 1.00
## y_rep[187]  149.58    0.13 5.70  142.19  149.58  156.99  1992 1.00
## y_rep[188]  160.32    0.12 5.52  153.36  160.20  167.67  1983 1.00
## y_rep[189]  149.74    0.12 5.54  142.59  149.79  156.83  2031 1.00
## y_rep[190]  160.42    0.13 5.71  153.03  160.54  167.70  1826 1.00
## y_rep[191]  149.57    0.13 5.62  142.39  149.53  156.64  1937 1.00
## y_rep[192]  160.53    0.13 5.47  153.72  160.36  167.49  1791 1.00
## y_rep[193]  149.44    0.13 5.53  142.37  149.45  156.39  1878 1.00
## y_rep[194]  160.35    0.13 5.51  153.44  160.36  167.26  1815 1.00
## y_rep[195]  149.33    0.12 5.47  142.62  149.21  156.58  2149 1.00
## y_rep[196]  149.38    0.12 5.61  142.13  149.43  156.65  2108 1.00
## y_rep[197]  149.52    0.13 5.64  142.15  149.49  156.97  1923 1.00
## y_rep[198]  160.23    0.13 5.70  153.00  160.27  167.57  1936 1.00
## y_rep[199]  160.26    0.12 5.39  153.32  160.22  166.95  2031 1.00
## y_rep[200]  149.59    0.12 5.44  142.27  149.80  156.48  1897 1.00
## y_rep[201]  160.49    0.13 5.54  153.40  160.34  167.49  1896 1.00
## y_rep[202]  149.37    0.12 5.51  142.30  149.34  156.59  2003 1.00
## y_rep[203]  160.36    0.12 5.61  152.97  160.51  167.24  2086 1.00
## y_rep[204]  149.46    0.13 5.47  142.44  149.53  156.69  1758 1.00
## y_rep[205]  160.41    0.13 5.49  153.24  160.47  167.32  1885 1.00
## y_rep[206]  149.42    0.13 5.62  142.34  149.43  156.59  1974 1.00
## y_rep[207]  149.78    0.13 5.60  142.60  149.82  156.95  1906 1.00
## y_rep[208]  149.48    0.13 5.49  142.60  149.38  156.62  1823 1.00
## y_rep[209]  149.59    0.13 5.52  142.60  149.86  156.31  1813 1.00
## y_rep[210]  149.73    0.14 5.61  142.42  149.81  156.89  1706 1.00
## y_rep[211]  160.28    0.13 5.60  153.18  160.18  167.73  1825 1.00
## y_rep[212]  160.23    0.13 5.63  153.13  160.21  167.37  1903 1.00
## y_rep[213]  160.60    0.13 5.74  153.36  160.58  168.05  1983 1.00
## y_rep[214]  149.53    0.14 5.62  142.23  149.44  156.97  1722 1.00
## y_rep[215]  160.28    0.13 5.57  152.99  160.37  167.24  1920 1.00
## y_rep[216]  160.41    0.13 5.76  153.13  160.35  167.70  1958 1.00
## y_rep[217]  160.35    0.13 5.59  153.38  160.31  167.49  1918 1.00
## y_rep[218]  149.75    0.12 5.57  142.65  149.80  156.81  2018 1.00
## y_rep[219]  160.39    0.13 5.41  153.26  160.52  167.13  1682 1.00
## y_rep[220]  149.62    0.13 5.70  142.34  149.60  157.02  1911 1.00
## y_rep[221]  149.44    0.12 5.78  142.10  149.50  156.78  2181 1.00
## y_rep[222]  149.76    0.12 5.44  143.01  149.70  156.83  2000 1.00
## y_rep[223]  160.31    0.12 5.60  152.82  160.43  167.36  2085 1.00
## y_rep[224]  149.44    0.12 5.66  142.49  149.44  156.72  2094 1.00
## y_rep[225]  160.28    0.13 5.60  152.96  160.32  167.42  1964 1.00
## y_rep[226]  149.50    0.12 5.61  142.59  149.41  156.83  2122 1.00
## y_rep[227]  149.27    0.13 5.60  142.17  149.31  156.41  1899 1.00
## y_rep[228]  160.39    0.13 5.78  152.83  160.39  167.93  1953 1.00
## y_rep[229]  149.38    0.13 5.37  142.44  149.38  156.36  1821 1.00
## y_rep[230]  160.43    0.12 5.53  153.31  160.24  167.55  2047 1.00
## y_rep[231]  160.45    0.12 5.61  153.49  160.38  167.67  2036 1.00
## y_rep[232]  149.43    0.12 5.45  142.46  149.39  156.47  1945 1.00
## y_rep[233]  149.63    0.13 5.62  142.36  149.66  156.87  2019 1.00
## y_rep[234]  160.32    0.12 5.56  153.06  160.48  167.31  1995 1.00
## y_rep[235]  160.38    0.13 5.66  153.09  160.40  167.55  1890 1.00
## y_rep[236]  149.62    0.13 5.50  142.68  149.56  156.74  1927 1.00
## y_rep[237]  160.30    0.12 5.55  153.06  160.39  167.34  2107 1.00
## y_rep[238]  149.60    0.13 5.45  142.56  149.63  156.61  1898 1.00
## y_rep[239]  149.43    0.12 5.46  142.22  149.51  156.35  1950 1.00
## y_rep[240]  149.68    0.13 5.57  142.57  149.61  156.84  1957 1.00
## y_rep[241]  149.65    0.13 5.55  142.41  149.65  156.84  1872 1.00
## y_rep[242]  160.25    0.13 5.60  153.08  160.21  167.45  1887 1.00
## y_rep[243]  160.10    0.12 5.48  153.17  160.11  167.16  2222 1.00
## y_rep[244]  149.39    0.13 5.55  142.32  149.28  156.51  1950 1.00
## y_rep[245]  160.33    0.13 5.62  153.37  160.21  167.48  1952 1.00
## y_rep[246]  149.50    0.13 5.55  142.13  149.54  156.55  1946 1.00
## y_rep[247]  160.36    0.13 5.63  153.12  160.19  167.72  2029 1.00
## y_rep[248]  149.51    0.13 5.67  142.44  149.51  156.87  1967 1.00
## y_rep[249]  160.46    0.13 5.67  153.11  160.30  167.91  2030 1.00
## y_rep[250]  160.36    0.13 5.76  152.99  160.41  167.49  1961 1.00
## y_rep[251]  160.48    0.12 5.52  153.48  160.42  167.55  2096 1.00
## y_rep[252]  160.39    0.12 5.61  153.24  160.44  167.60  2136 1.00
## y_rep[253]  149.62    0.13 5.54  142.76  149.52  156.56  1864 1.00
## y_rep[254]  149.49    0.13 5.65  142.28  149.47  156.62  2039 1.00
## y_rep[255]  149.55    0.12 5.41  142.55  149.50  156.51  2104 1.00
## y_rep[256]  149.55    0.13 5.46  142.63  149.52  156.49  1647 1.00
## y_rep[257]  160.19    0.13 5.57  153.09  160.13  167.45  1943 1.00
## y_rep[258]  149.44    0.12 5.57  142.40  149.49  156.43  1997 1.00
## y_rep[259]  160.42    0.12 5.48  153.36  160.40  167.44  2104 1.00
## y_rep[260]  160.40    0.12 5.52  153.49  160.33  167.20  1949 1.00
## y_rep[261]  160.40    0.12 5.60  153.18  160.32  167.62  2064 1.00
## y_rep[262]  149.58    0.12 5.31  142.48  149.51  156.39  2002 1.00
## y_rep[263]  149.47    0.13 5.51  142.41  149.42  156.53  1886 1.00
## y_rep[264]  149.48    0.12 5.61  142.26  149.60  156.47  2131 1.00
## y_rep[265]  160.49    0.12 5.68  153.18  160.53  167.72  2191 1.00
## y_rep[266]  149.58    0.13 5.63  142.49  149.55  156.78  1967 1.00
## y_rep[267]  160.22    0.13 5.71  153.06  160.16  167.38  1888 1.00
## y_rep[268]  149.41    0.13 5.58  142.24  149.38  156.64  1860 1.00
## y_rep[269]  149.57    0.12 5.66  142.21  149.75  156.52  2075 1.00
## y_rep[270]  160.26    0.12 5.46  153.48  160.19  167.34  2000 1.00
## y_rep[271]  160.67    0.12 5.42  153.90  160.71  167.64  2016 1.00
## y_rep[272]  149.57    0.12 5.68  142.43  149.47  156.89  2119 1.00
## y_rep[273]  160.02    0.13 5.58  152.57  160.14  166.98  1906 1.00
## y_rep[274]  160.32    0.13 5.56  153.16  160.26  167.56  1929 1.00
## y_rep[275]  149.54    0.12 5.61  142.49  149.43  156.69  2062 1.00
## y_rep[276]  160.36    0.13 5.62  152.85  160.46  167.57  1923 1.00
## y_rep[277]  149.40    0.13 5.45  142.27  149.51  156.33  1775 1.00
## y_rep[278]  149.53    0.13 5.50  142.49  149.51  156.59  1677 1.00
## y_rep[279]  160.41    0.13 5.63  153.16  160.38  167.59  1951 1.00
## y_rep[280]  149.45    0.12 5.50  142.40  149.54  156.46  1972 1.00
## y_rep[281]  149.39    0.13 5.64  142.32  149.43  156.64  1954 1.00
## y_rep[282]  160.43    0.12 5.46  153.14  160.53  167.40  2203 1.00
## y_rep[283]  149.69    0.12 5.50  142.74  149.65  156.59  2158 1.00
## y_rep[284]  160.38    0.13 5.44  153.46  160.37  167.30  1887 1.00
## y_rep[285]  149.50    0.13 5.50  142.55  149.58  156.48  1876 1.00
## y_rep[286]  160.39    0.13 5.66  153.05  160.51  167.61  1986 1.00
## y_rep[287]  160.46    0.13 5.54  153.35  160.46  167.45  1931 1.00
## y_rep[288]  160.47    0.13 5.57  153.41  160.73  167.28  1898 1.00
## y_rep[289]  149.60    0.12 5.43  142.60  149.62  156.44  1981 1.00
## y_rep[290]  160.19    0.13 5.50  153.14  160.32  166.97  1925 1.00
## y_rep[291]  160.37    0.12 5.51  153.47  160.18  167.44  1989 1.00
## y_rep[292]  160.41    0.12 5.63  153.21  160.47  167.81  2210 1.00
## y_rep[293]  160.59    0.13 5.69  153.08  160.47  167.96  2011 1.00
## y_rep[294]  149.71    0.12 5.60  142.72  149.71  156.96  2016 1.00
## y_rep[295]  160.32    0.12 5.44  153.07  160.53  167.14  2000 1.00
## y_rep[296]  149.40    0.12 5.50  142.47  149.23  156.62  1977 1.00
## y_rep[297]  149.47    0.14 5.69  142.15  149.43  156.77  1610 1.00
## y_rep[298]  149.32    0.13 5.54  142.13  149.31  156.40  1940 1.00
## y_rep[299]  160.51    0.13 5.55  153.43  160.38  167.76  1947 1.00
## y_rep[300]  160.55    0.14 5.60  153.52  160.48  167.86  1668 1.00
## y_rep[301]  149.61    0.12 5.56  142.26  149.67  156.70  2029 1.00
## y_rep[302]  160.17    0.13 5.47  153.12  160.24  166.92  1840 1.00
## y_rep[303]  149.41    0.13 5.58  142.03  149.36  156.43  1843 1.00
## y_rep[304]  160.36    0.14 5.71  153.03  160.38  167.56  1740 1.00
## y_rep[305]  160.38    0.12 5.54  153.28  160.24  167.62  2050 1.00
## y_rep[306]  160.34    0.13 5.64  153.21  160.32  167.48  1756 1.00
## y_rep[307]  160.45    0.12 5.56  153.53  160.39  167.53  2170 1.00
## y_rep[308]  149.52    0.14 5.74  142.28  149.56  156.84  1749 1.00
## y_rep[309]  149.50    0.12 5.55  142.54  149.53  156.76  2010 1.00
## y_rep[310]  149.49    0.13 5.46  142.69  149.34  156.68  1829 1.00
## y_rep[311]  149.34    0.14 5.68  142.31  149.42  156.69  1605 1.00
## y_rep[312]  149.56    0.12 5.55  142.23  149.51  156.67  1973 1.00
## y_rep[313]  160.48    0.12 5.51  153.61  160.38  167.53  2170 1.00
## y_rep[314]  149.48    0.13 5.60  142.41  149.49  156.60  1998 1.00
## y_rep[315]  160.33    0.13 5.60  153.17  160.37  167.50  1894 1.00
## y_rep[316]  149.41    0.12 5.70  141.95  149.32  156.75  2094 1.00
## y_rep[317]  160.37    0.13 5.59  153.06  160.38  167.56  1912 1.00
## y_rep[318]  149.39    0.13 5.63  142.22  149.33  156.74  1825 1.00
## y_rep[319]  160.20    0.13 5.62  152.97  160.12  167.49  1804 1.00
## y_rep[320]  149.46    0.13 5.52  142.31  149.49  156.43  1903 1.00
## y_rep[321]  160.49    0.13 5.54  153.53  160.55  167.43  1864 1.00
## y_rep[322]  149.57    0.12 5.36  142.64  149.67  156.22  1924 1.00
## y_rep[323]  160.36    0.14 5.49  153.29  160.31  167.21  1588 1.00
## y_rep[324]  160.47    0.13 5.50  153.55  160.42  167.56  1913 1.00
## y_rep[325]  149.64    0.13 5.70  142.34  149.67  156.87  1791 1.00
## y_rep[326]  149.77    0.13 5.66  142.21  149.92  156.84  1962 1.00
## y_rep[327]  149.62    0.12 5.58  142.37  149.59  156.87  2166 1.00
## y_rep[328]  160.12    0.13 5.51  152.99  160.14  167.20  1726 1.00
## y_rep[329]  160.36    0.13 5.64  152.95  160.33  167.58  1900 1.00
## y_rep[330]  160.36    0.13 5.75  153.05  160.40  167.93  1990 1.00
## y_rep[331]  160.25    0.13 5.61  153.09  160.29  167.46  2002 1.00
## y_rep[332]  160.25    0.13 5.60  153.09  160.12  167.59  1866 1.00
## y_rep[333]  149.43    0.13 5.70  142.21  149.44  156.79  1879 1.00
## y_rep[334]  160.49    0.12 5.40  153.50  160.42  167.53  1961 1.00
## y_rep[335]  149.63    0.12 5.55  142.45  149.63  156.66  1986 1.00
## y_rep[336]  160.41    0.13 5.52  153.38  160.38  167.52  1864 1.00
## y_rep[337]  149.65    0.13 5.68  142.41  149.60  156.95  2048 1.00
## y_rep[338]  149.20    0.13 5.59  141.71  149.24  156.50  1924 1.00
## y_rep[339]  149.55    0.13 5.65  142.58  149.53  156.68  1801 1.00
## y_rep[340]  149.57    0.12 5.48  142.57  149.53  156.55  2013 1.00
## y_rep[341]  149.49    0.12 5.52  142.34  149.55  156.62  2019 1.00
## y_rep[342]  160.27    0.13 5.52  153.30  160.25  167.38  1797 1.00
## y_rep[343]  149.34    0.13 5.63  142.14  149.36  156.30  1954 1.00
## y_rep[344]  160.33    0.13 5.66  153.21  160.32  167.54  1867 1.00
## y_rep[345]  149.74    0.12 5.42  142.98  149.62  156.83  1979 1.00
## y_rep[346]  160.32    0.12 5.45  153.19  160.32  167.21  1908 1.00
## y_rep[347]  149.46    0.14 5.59  142.19  149.45  156.86  1649 1.00
## y_rep[348]  149.65    0.13 5.57  142.42  149.61  156.93  1861 1.00
## y_rep[349]  149.47    0.13 5.56  142.43  149.46  156.60  1978 1.00
## y_rep[350]  160.22    0.14 5.45  153.38  160.09  167.26  1589 1.00
## y_rep[351]  149.60    0.12 5.45  142.44  149.57  156.65  1914 1.00
## y_rep[352]  160.49    0.13 5.56  153.09  160.68  167.30  1937 1.00
## lp__       -780.37    0.04 1.11 -781.84 -780.08 -779.27   673 1.00
## 
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:53:34 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).
summary(lm(height~as.factor(male),data=howell2))
## 
## Call:
## lm(formula = height ~ as.factor(male), data = howell2)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -20.6585  -3.4635   0.2965   3.4715  18.7115 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      149.5135     0.4049  369.25   <2e-16 ***
## as.factor(male)1  10.8450     0.5914   18.34   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.537 on 350 degrees of freedom
## Multiple R-squared:   0.49,  Adjusted R-squared:  0.4885 
## F-statistic: 336.3 on 1 and 350 DF,  p-value: < 2.2e-16

Convergence diagnostics

posterior <- as.array(fit.m5_1_1) # posterior draws
lp <- log_posterior(fit.m5_1_1) # log of the posterior density for each draw 
np <- nuts_params(fit.m5_1_1) # Some NUTS-specific diagnostic values 
mcmc_parcoord(posterior, np = np,pars=c("beta[1]","beta[2]","sigma")) 

mcmc_pairs(posterior, np = np, pars = c("beta[1]","beta[2]","sigma"),  off_diag_args = list(size = 0.75)) # Pairs plots

mcmc_trace(posterior, pars = c("beta[1]","beta[2]","sigma"), np = np) +  xlab("Post-warmup iteration") # Trace plots
## No divergences to plot.

Same outputs with another prior for sigma: \(\sigma \sim \, {\rm Cauchy} (0,5)\)

data {                                                     
  int<lower=0> N;                                          // number of observations
  int<lower=0> n;                                          // number of levels of the categorical variable
  vector[N] height;                                        // response variables
  matrix[N,n] X;                                           // categorical variable
}

parameters {                                          
  vector[n] beta;                                         // intercepts of the categorical variable
  real sigma;
}

transformed parameters {
vector[N] mu;                                             
mu = X*beta;  
}

model {
  height ~ normal(mu ,sigma);                              // likelihood
  sigma ~ cauchy(0,5);
  beta[1] ~ normal(178,20);
  beta[2] ~ normal(0,10);
  // beta ~ normal(0,1000)
}

generated quantities {
  vector[N] y_rep;
 for(i in 1:N)
 y_rep[i] = normal_rng(mu[i], sigma);
}
fit.m5_1_2 <- sampling(m5.1.2, data = m5.1.data, iter = 1000, chains = 4, cores = 4)
print(fit.m5_1_2, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: 4c192013905ef0bcb3999ba4bc93cd6e.
## 4 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=2000.
## 
##               mean se_mean   sd     10%     50%     90% n_eff Rhat
## beta[1]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## beta[2]      10.78    0.02 0.60   10.03   10.77   11.54  1292    1
## sigma         5.54    0.01 0.21    5.26    5.53    5.81  1498    1
## mu[1]       160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[2]       149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[3]       149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[4]       160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[5]       149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[6]       160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[7]       149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[8]       160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[9]       149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[10]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[11]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[12]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[13]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[14]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[15]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[16]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[17]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[18]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[19]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[20]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[21]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[22]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[23]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[24]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[25]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[26]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[27]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[28]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[29]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[30]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[31]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[32]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[33]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[34]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[35]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[36]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[37]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[38]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[39]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[40]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[41]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[42]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[43]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[44]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[45]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[46]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[47]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[48]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[49]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[50]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[51]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[52]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[53]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[54]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[55]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[56]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[57]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[58]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[59]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[60]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[61]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[62]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[63]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[64]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[65]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[66]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[67]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[68]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[69]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[70]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[71]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[72]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[73]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[74]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[75]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[76]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[77]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[78]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[79]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[80]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[81]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[82]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[83]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[84]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[85]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[86]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[87]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[88]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[89]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[90]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[91]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[92]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[93]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[94]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[95]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[96]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[97]      149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[98]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[99]      160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[100]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[101]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[102]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[103]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[104]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[105]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[106]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[107]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[108]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[109]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[110]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[111]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[112]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[113]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[114]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[115]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[116]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[117]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[118]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[119]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[120]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[121]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[122]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[123]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[124]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[125]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[126]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[127]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[128]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[129]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[130]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[131]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[132]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[133]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[134]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[135]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[136]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[137]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[138]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[139]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[140]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[141]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[142]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[143]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[144]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[145]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[146]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[147]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[148]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[149]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[150]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[151]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[152]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[153]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[154]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[155]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[156]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[157]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[158]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[159]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[160]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[161]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[162]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[163]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[164]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[165]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[166]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[167]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[168]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[169]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[170]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[171]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[172]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[173]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[174]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[175]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[176]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[177]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[178]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[179]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[180]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[181]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[182]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[183]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[184]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[185]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[186]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[187]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[188]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[189]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[190]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[191]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[192]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[193]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[194]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[195]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[196]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[197]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[198]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[199]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[200]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[201]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[202]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[203]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[204]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[205]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[206]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[207]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[208]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[209]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[210]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[211]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[212]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[213]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[214]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[215]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[216]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[217]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[218]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[219]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[220]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[221]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[222]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[223]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[224]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[225]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[226]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[227]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[228]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[229]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[230]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[231]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[232]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[233]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[234]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[235]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[236]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[237]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[238]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[239]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[240]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[241]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[242]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[243]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[244]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[245]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[246]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[247]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[248]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[249]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[250]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[251]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[252]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[253]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[254]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[255]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[256]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[257]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[258]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[259]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[260]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[261]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[262]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[263]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[264]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[265]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[266]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[267]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[268]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[269]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[270]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[271]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[272]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[273]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[274]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[275]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[276]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[277]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[278]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[279]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[280]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[281]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[282]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[283]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[284]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[285]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[286]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[287]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[288]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[289]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[290]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[291]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[292]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[293]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[294]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[295]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[296]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[297]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[298]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[299]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[300]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[301]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[302]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[303]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[304]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[305]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[306]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[307]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[308]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[309]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[310]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[311]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[312]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[313]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[314]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[315]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[316]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[317]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[318]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[319]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[320]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[321]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[322]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[323]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[324]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[325]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[326]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[327]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[328]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[329]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[330]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[331]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[332]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[333]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[334]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[335]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[336]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[337]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[338]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[339]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[340]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[341]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[342]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[343]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[344]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[345]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[346]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[347]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[348]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[349]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[350]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## mu[351]     149.55    0.01 0.41  149.03  149.55  150.05  1204    1
## mu[352]     160.33    0.01 0.40  159.83  160.33  160.84  2237    1
## y_rep[1]    160.13    0.12 5.59  153.00  160.17  167.24  2073    1
## y_rep[2]    149.77    0.12 5.47  142.76  149.78  156.81  1958    1
## y_rep[3]    149.47    0.12 5.44  142.22  149.53  156.48  1976    1
## y_rep[4]    160.44    0.13 5.70  153.07  160.37  167.93  2051    1
## y_rep[5]    149.75    0.13 5.47  142.70  149.79  156.65  1917    1
## y_rep[6]    160.41    0.12 5.67  153.22  160.39  167.56  2062    1
## y_rep[7]    149.55    0.13 5.40  142.60  149.54  156.47  1800    1
## y_rep[8]    160.45    0.13 5.49  153.62  160.31  167.53  1828    1
## y_rep[9]    149.71    0.14 5.61  142.38  149.72  156.88  1664    1
## y_rep[10]   160.60    0.13 5.71  153.47  160.49  167.83  2067    1
## y_rep[11]   149.55    0.13 5.46  142.45  149.51  156.66  1788    1
## y_rep[12]   160.24    0.12 5.63  152.97  160.31  167.36  2045    1
## y_rep[13]   149.80    0.12 5.35  143.11  150.04  156.64  2064    1
## y_rep[14]   149.32    0.13 5.47  142.29  149.32  156.23  1792    1
## y_rep[15]   149.57    0.12 5.45  142.37  149.59  156.61  1907    1
## y_rep[16]   160.21    0.12 5.57  152.97  160.23  167.31  2010    1
## y_rep[17]   160.16    0.12 5.40  153.21  160.34  166.83  2071    1
## y_rep[18]   149.48    0.13 5.47  142.68  149.34  156.74  1834    1
## y_rep[19]   160.29    0.12 5.59  153.06  160.28  167.45  2043    1
## y_rep[20]   149.44    0.13 5.49  142.48  149.36  156.68  1677    1
## y_rep[21]   160.53    0.13 5.70  153.20  160.61  167.74  2000    1
## y_rep[22]   149.76    0.12 5.39  142.96  149.74  156.67  2057    1
## y_rep[23]   149.69    0.12 5.56  142.42  149.77  156.89  2020    1
## y_rep[24]   160.57    0.12 5.51  153.44  160.57  167.46  1965    1
## y_rep[25]   160.25    0.12 5.41  153.34  160.32  167.28  1982    1
## y_rep[26]   149.69    0.13 5.60  142.48  149.67  156.66  1866    1
## y_rep[27]   149.47    0.13 5.48  142.36  149.58  156.38  1740    1
## y_rep[28]   149.68    0.13 5.59  142.53  149.63  156.77  1733    1
## y_rep[29]   149.78    0.13 5.59  142.46  149.67  156.97  1896    1
## y_rep[30]   149.55    0.12 5.37  142.66  149.64  156.36  2113    1
## y_rep[31]   149.38    0.13 5.52  142.33  149.33  156.43  1807    1
## y_rep[32]   160.32    0.13 5.55  153.33  160.13  167.45  1801    1
## y_rep[33]   160.37    0.13 5.57  153.09  160.50  167.06  1750    1
## y_rep[34]   160.41    0.13 5.78  153.05  160.43  167.79  2103    1
## y_rep[35]   160.12    0.13 5.51  153.10  159.98  167.12  1888    1
## y_rep[36]   149.40    0.13 5.62  142.26  149.31  156.66  1833    1
## y_rep[37]   149.62    0.12 5.49  142.77  149.59  156.62  2163    1
## y_rep[38]   160.48    0.13 5.52  153.27  160.52  167.57  1927    1
## y_rep[39]   149.72    0.12 5.61  142.44  149.61  156.96  2058    1
## y_rep[40]   160.40    0.13 5.66  152.96  160.21  167.70  1952    1
## y_rep[41]   160.21    0.12 5.70  153.16  160.19  167.15  2132    1
## y_rep[42]   160.38    0.13 5.56  153.13  160.34  167.44  1886    1
## y_rep[43]   149.38    0.13 5.72  142.08  149.36  156.43  2017    1
## y_rep[44]   149.39    0.13 5.43  142.24  149.55  156.25  1854    1
## y_rep[45]   149.70    0.12 5.57  142.74  149.62  156.82  1996    1
## y_rep[46]   149.58    0.13 5.68  142.32  149.60  156.83  1865    1
## y_rep[47]   160.25    0.12 5.47  153.23  160.27  167.40  2066    1
## y_rep[48]   149.57    0.12 5.55  142.65  149.53  156.83  1982    1
## y_rep[49]   160.15    0.12 5.61  153.15  160.21  167.38  2031    1
## y_rep[50]   149.57    0.13 5.65  142.12  149.55  156.73  1889    1
## y_rep[51]   149.72    0.13 5.67  142.39  149.62  157.03  1768    1
## y_rep[52]   149.28    0.13 5.66  142.06  149.22  156.52  1903    1
## y_rep[53]   160.22    0.12 5.49  153.23  160.32  167.14  2028    1
## y_rep[54]   160.13    0.12 5.72  152.86  159.98  167.63  2227    1
## y_rep[55]   149.60    0.12 5.43  142.51  149.54  156.72  2002    1
## y_rep[56]   160.34    0.13 5.55  153.11  160.44  167.64  1969    1
## y_rep[57]   149.53    0.12 5.48  142.56  149.39  156.63  2210    1
## y_rep[58]   160.40    0.12 5.52  153.31  160.43  167.44  2021    1
## y_rep[59]   149.63    0.12 5.50  142.29  149.71  156.61  2144    1
## y_rep[60]   160.47    0.13 5.62  153.28  160.49  167.50  1858    1
## y_rep[61]   149.72    0.13 5.71  142.34  149.82  156.86  1969    1
## y_rep[62]   149.60    0.12 5.56  142.49  149.59  156.78  2180    1
## y_rep[63]   160.31    0.12 5.62  153.16  160.43  167.26  2129    1
## y_rep[64]   160.30    0.12 5.47  153.40  160.16  167.42  2240    1
## y_rep[65]   149.59    0.13 5.72  142.13  149.70  156.88  1925    1
## y_rep[66]   160.24    0.13 5.55  153.35  160.36  167.27  1867    1
## y_rep[67]   149.56    0.13 5.64  142.19  149.42  156.84  1986    1
## y_rep[68]   149.55    0.12 5.47  142.68  149.50  156.65  1995    1
## y_rep[69]   160.47    0.15 5.49  153.39  160.51  167.32  1392    1
## y_rep[70]   149.56    0.13 5.67  142.38  149.53  156.87  2041    1
## y_rep[71]   160.26    0.13 5.58  153.02  160.18  167.33  1952    1
## y_rep[72]   160.46    0.13 5.67  153.24  160.38  167.79  1850    1
## y_rep[73]   160.26    0.13 5.50  153.17  160.21  167.25  1797    1
## y_rep[74]   149.49    0.14 5.59  142.29  149.48  156.84  1620    1
## y_rep[75]   149.50    0.12 5.58  142.67  149.37  156.61  2069    1
## y_rep[76]   149.43    0.12 5.52  142.43  149.41  156.35  2103    1
## y_rep[77]   149.69    0.12 5.53  142.61  149.69  156.74  2211    1
## y_rep[78]   160.41    0.12 5.40  153.54  160.37  167.32  1893    1
## y_rep[79]   149.60    0.14 5.57  142.69  149.58  156.75  1685    1
## y_rep[80]   149.65    0.13 5.41  142.95  149.47  156.81  1871    1
## y_rep[81]   160.21    0.13 5.65  152.98  160.27  167.52  1855    1
## y_rep[82]   160.42    0.13 5.61  153.17  160.28  167.54  1935    1
## y_rep[83]   149.33    0.12 5.43  142.22  149.41  156.38  1920    1
## y_rep[84]   160.28    0.13 5.60  153.16  160.20  167.38  1998    1
## y_rep[85]   149.62    0.14 5.43  142.91  149.67  156.56  1523    1
## y_rep[86]   160.49    0.13 5.61  153.41  160.55  167.89  1982    1
## y_rep[87]   149.68    0.14 5.71  142.33  149.66  157.10  1714    1
## y_rep[88]   160.38    0.13 5.59  153.19  160.27  167.60  1967    1
## y_rep[89]   149.38    0.13 5.64  142.20  149.36  156.54  1963    1
## y_rep[90]   160.27    0.12 5.58  153.30  160.07  167.65  2092    1
## y_rep[91]   160.33    0.12 5.50  153.46  160.41  167.39  2094    1
## y_rep[92]   160.49    0.12 5.40  153.56  160.47  167.36  1978    1
## y_rep[93]   149.61    0.13 5.53  142.57  149.57  156.82  1949    1
## y_rep[94]   149.48    0.12 5.55  142.27  149.63  156.61  1991    1
## y_rep[95]   160.28    0.13 5.56  153.40  160.37  167.24  1961    1
## y_rep[96]   160.27    0.13 5.58  153.10  160.26  167.15  1755    1
## y_rep[97]   149.49    0.13 5.55  142.15  149.65  156.68  1800    1
## y_rep[98]   160.56    0.13 5.59  153.48  160.62  167.77  1958    1
## y_rep[99]   160.19    0.12 5.45  153.38  160.19  167.19  1966    1
## y_rep[100]  149.51    0.12 5.44  142.69  149.36  156.43  1986    1
## y_rep[101]  160.30    0.13 5.48  153.50  160.27  167.33  1915    1
## y_rep[102]  149.58    0.13 5.54  142.66  149.41  156.97  1927    1
## y_rep[103]  149.70    0.12 5.37  142.99  149.58  156.59  2012    1
## y_rep[104]  160.35    0.12 5.67  152.91  160.41  167.49  2074    1
## y_rep[105]  160.30    0.13 5.62  153.15  160.40  167.29  1932    1
## y_rep[106]  149.47    0.13 5.55  142.46  149.44  156.74  1853    1
## y_rep[107]  149.56    0.13 5.61  142.07  149.65  156.62  1989    1
## y_rep[108]  149.51    0.13 5.65  142.26  149.64  156.76  2022    1
## y_rep[109]  160.32    0.12 5.56  153.19  160.22  167.50  1983    1
## y_rep[110]  149.60    0.13 5.65  142.38  149.57  156.70  1958    1
## y_rep[111]  160.20    0.12 5.51  153.04  160.31  167.33  1993    1
## y_rep[112]  149.62    0.13 5.60  142.29  149.72  156.69  1956    1
## y_rep[113]  149.49    0.12 5.69  142.16  149.49  156.93  2073    1
## y_rep[114]  160.25    0.12 5.52  153.19  160.16  167.36  2014    1
## y_rep[115]  160.35    0.13 5.64  153.34  160.38  167.42  1957    1
## y_rep[116]  149.54    0.12 5.53  142.46  149.50  156.64  1962    1
## y_rep[117]  160.27    0.14 5.67  153.04  160.26  167.77  1662    1
## y_rep[118]  160.31    0.13 5.57  153.04  160.52  167.15  1821    1
## y_rep[119]  149.74    0.13 5.56  142.81  149.77  156.75  1827    1
## y_rep[120]  149.62    0.13 5.64  142.38  149.67  156.99  2017    1
## y_rep[121]  160.21    0.13 5.67  152.95  160.27  167.31  1989    1
## y_rep[122]  149.51    0.12 5.40  142.55  149.41  156.49  1920    1
## y_rep[123]  160.60    0.13 5.56  153.56  160.57  167.66  1818    1
## y_rep[124]  149.53    0.12 5.47  142.64  149.27  156.68  1939    1
## y_rep[125]  149.62    0.13 5.64  142.24  149.74  156.56  2011    1
## y_rep[126]  149.51    0.13 5.62  142.43  149.45  156.74  1986    1
## y_rep[127]  149.51    0.13 5.76  142.06  149.48  156.97  1878    1
## y_rep[128]  160.15    0.13 5.59  153.13  160.19  167.34  1795    1
## y_rep[129]  149.54    0.12 5.40  142.71  149.43  156.50  1929    1
## y_rep[130]  149.41    0.12 5.49  142.34  149.29  156.51  1952    1
## y_rep[131]  160.56    0.14 5.63  153.17  160.65  167.95  1719    1
## y_rep[132]  160.23    0.13 5.49  153.05  160.37  167.23  1849    1
## y_rep[133]  149.50    0.13 5.63  142.37  149.42  156.71  2019    1
## y_rep[134]  160.37    0.12 5.59  153.32  160.37  167.33  2008    1
## y_rep[135]  149.51    0.13 5.66  142.31  149.41  156.71  2009    1
## y_rep[136]  160.04    0.13 5.75  152.61  159.99  167.54  2001    1
## y_rep[137]  160.29    0.14 5.66  153.28  160.07  167.68  1708    1
## y_rep[138]  160.20    0.13 5.60  153.06  160.09  167.44  1929    1
## y_rep[139]  149.55    0.12 5.51  142.45  149.61  156.57  2040    1
## y_rep[140]  160.54    0.13 5.55  153.42  160.41  167.68  1868    1
## y_rep[141]  149.65    0.13 5.57  142.53  149.68  156.65  1803    1
## y_rep[142]  149.58    0.13 5.54  142.40  149.64  156.77  1863    1
## y_rep[143]  149.56    0.13 5.64  142.38  149.65  156.70  1877    1
## y_rep[144]  149.61    0.12 5.57  142.53  149.47  156.85  2104    1
## y_rep[145]  149.53    0.12 5.49  142.51  149.51  156.74  2035    1
## y_rep[146]  160.03    0.13 5.54  152.92  160.02  167.24  1930    1
## y_rep[147]  149.68    0.13 5.57  142.68  149.66  156.94  1921    1
## y_rep[148]  149.54    0.13 5.51  142.72  149.45  156.66  1919    1
## y_rep[149]  160.17    0.12 5.42  153.31  160.13  166.93  2060    1
## y_rep[150]  149.53    0.13 5.70  142.41  149.53  156.95  2020    1
## y_rep[151]  160.42    0.13 5.59  153.52  160.35  167.54  1862    1
## y_rep[152]  149.52    0.13 5.50  142.28  149.57  156.47  1908    1
## y_rep[153]  160.16    0.13 5.66  152.99  160.29  167.34  1770    1
## y_rep[154]  149.68    0.13 5.60  142.37  149.72  156.75  1945    1
## y_rep[155]  160.45    0.13 5.53  153.34  160.50  167.39  1950    1
## y_rep[156]  149.55    0.13 5.73  142.00  149.54  157.01  1884    1
## y_rep[157]  149.38    0.12 5.42  142.39  149.43  156.40  2023    1
## y_rep[158]  160.12    0.13 5.57  153.24  160.00  167.14  1955    1
## y_rep[159]  160.34    0.12 5.58  153.21  160.38  167.54  2051    1
## y_rep[160]  149.57    0.13 5.62  142.34  149.73  157.00  1832    1
## y_rep[161]  160.21    0.13 5.61  153.21  160.24  167.25  1970    1
## y_rep[162]  149.43    0.13 5.54  141.98  149.48  156.43  1875    1
## y_rep[163]  160.45    0.13 5.56  153.18  160.40  167.76  1824    1
## y_rep[164]  160.31    0.13 5.58  153.05  160.26  167.68  1990    1
## y_rep[165]  160.34    0.13 5.67  153.11  160.39  167.71  1959    1
## y_rep[166]  149.52    0.13 5.57  142.29  149.48  156.43  1950    1
## y_rep[167]  160.25    0.13 5.61  153.27  160.25  167.33  1871    1
## y_rep[168]  149.68    0.12 5.55  142.89  149.55  156.83  2049    1
## y_rep[169]  149.49    0.13 5.47  142.42  149.51  156.45  1864    1
## y_rep[170]  149.66    0.12 5.54  142.37  149.77  156.64  2190    1
## y_rep[171]  149.62    0.12 5.39  142.63  149.52  156.64  1905    1
## y_rep[172]  160.54    0.14 5.65  153.15  160.53  167.81  1718    1
## y_rep[173]  149.76    0.12 5.48  142.50  149.68  156.83  1979    1
## y_rep[174]  160.44    0.13 5.54  153.49  160.45  167.56  1802    1
## y_rep[175]  149.61    0.12 5.56  142.67  149.53  156.74  2060    1
## y_rep[176]  160.22    0.12 5.66  153.17  160.03  167.42  2135    1
## y_rep[177]  149.67    0.13 5.61  142.38  149.53  156.90  2011    1
## y_rep[178]  149.63    0.12 5.46  142.57  149.70  156.51  1921    1
## y_rep[179]  160.14    0.13 5.60  153.10  160.11  167.33  1986    1
## y_rep[180]  149.66    0.12 5.51  142.62  149.70  156.49  2119    1
## y_rep[181]  159.97    0.12 5.43  153.19  159.85  166.96  1910    1
## y_rep[182]  149.59    0.12 5.53  142.35  149.58  156.77  2118    1
## y_rep[183]  149.68    0.12 5.39  142.86  149.59  156.92  1878    1
## y_rep[184]  149.40    0.12 5.46  142.25  149.38  156.25  2180    1
## y_rep[185]  160.34    0.12 5.52  153.46  160.28  167.45  2015    1
## y_rep[186]  160.10    0.12 5.67  152.97  159.95  167.44  2055    1
## y_rep[187]  149.58    0.12 5.55  142.53  149.62  156.68  2156    1
## y_rep[188]  160.42    0.12 5.59  153.35  160.37  167.54  2019    1
## y_rep[189]  149.46    0.13 5.63  142.38  149.51  156.52  1872    1
## y_rep[190]  160.27    0.13 5.75  152.87  160.28  167.93  1918    1
## y_rep[191]  149.52    0.13 5.47  142.37  149.61  156.28  1863    1
## y_rep[192]  160.43    0.12 5.58  153.43  160.31  167.67  2031    1
## y_rep[193]  149.57    0.13 5.76  142.32  149.48  156.86  1994    1
## y_rep[194]  160.36    0.13 5.64  153.15  160.34  167.44  1937    1
## y_rep[195]  149.41    0.13 5.64  142.38  149.47  156.68  1881    1
## y_rep[196]  149.53    0.13 5.67  142.26  149.46  156.78  1977    1
## y_rep[197]  149.52    0.12 5.58  142.53  149.34  156.93  2009    1
## y_rep[198]  160.39    0.12 5.53  153.43  160.52  167.58  1961    1
## y_rep[199]  160.41    0.14 5.57  153.33  160.45  167.72  1672    1
## y_rep[200]  149.46    0.13 5.55  142.33  149.32  156.43  1959    1
## y_rep[201]  160.26    0.13 5.54  153.21  160.23  167.48  1767    1
## y_rep[202]  149.47    0.13 5.57  142.71  149.35  156.52  1969    1
## y_rep[203]  160.38    0.13 5.66  153.00  160.39  167.74  1882    1
## y_rep[204]  149.65    0.12 5.39  142.72  149.48  156.58  2011    1
## y_rep[205]  160.04    0.13 5.59  152.99  160.03  167.17  1959    1
## y_rep[206]  149.67    0.14 5.59  142.60  149.82  156.66  1690    1
## y_rep[207]  149.61    0.13 5.61  142.45  149.48  156.63  1846    1
## y_rep[208]  149.53    0.13 5.61  142.44  149.43  156.76  1881    1
## y_rep[209]  149.46    0.13 5.60  142.31  149.38  156.62  1938    1
## y_rep[210]  149.68    0.12 5.56  142.53  149.61  156.89  1991    1
## y_rep[211]  160.49    0.13 5.60  153.13  160.59  167.31  1785    1
## y_rep[212]  160.41    0.13 5.57  153.33  160.41  167.38  1961    1
## y_rep[213]  160.46    0.12 5.62  153.32  160.47  167.81  2078    1
## y_rep[214]  149.71    0.13 5.59  142.57  149.79  156.85  1986    1
## y_rep[215]  160.50    0.12 5.51  153.65  160.55  167.68  1984    1
## y_rep[216]  160.64    0.13 5.44  153.62  160.70  167.60  1720    1
## y_rep[217]  160.36    0.12 5.49  153.33  160.38  167.36  1974    1
## y_rep[218]  149.29    0.12 5.52  142.22  149.37  156.46  2036    1
## y_rep[219]  160.23    0.12 5.55  153.09  160.26  167.60  2015    1
## y_rep[220]  149.44    0.12 5.54  142.28  149.50  156.55  2132    1
## y_rep[221]  149.84    0.13 5.59  142.64  149.81  156.98  1985    1
## y_rep[222]  149.67    0.12 5.43  142.77  149.75  156.41  1913    1
## y_rep[223]  160.26    0.13 5.59  152.90  160.24  167.45  1795    1
## y_rep[224]  149.51    0.12 5.46  142.19  149.64  156.45  2041    1
## y_rep[225]  160.20    0.12 5.65  153.11  160.12  167.39  2062    1
## y_rep[226]  149.56    0.12 5.62  142.23  149.54  156.80  2066    1
## y_rep[227]  149.66    0.13 5.74  142.25  149.62  157.20  2023    1
## y_rep[228]  160.39    0.13 5.65  153.20  160.31  167.65  1994    1
## y_rep[229]  149.43    0.12 5.48  142.53  149.47  156.40  2059    1
## y_rep[230]  160.28    0.13 5.52  153.25  160.24  167.19  1822    1
## y_rep[231]  160.32    0.12 5.42  153.37  160.28  167.11  1964    1
## y_rep[232]  149.49    0.13 5.51  142.31  149.53  156.53  1897    1
## y_rep[233]  149.61    0.12 5.56  142.40  149.52  156.49  2011    1
## y_rep[234]  160.37    0.12 5.46  153.44  160.41  167.39  1941    1
## y_rep[235]  160.22    0.12 5.60  152.94  160.22  167.46  2010    1
## y_rep[236]  149.49    0.12 5.43  142.63  149.42  156.46  1894    1
## y_rep[237]  160.35    0.12 5.66  153.16  160.52  167.34  2121    1
## y_rep[238]  149.54    0.12 5.70  142.49  149.52  156.78  2114    1
## y_rep[239]  149.70    0.13 5.51  142.76  149.66  156.74  1914    1
## y_rep[240]  149.45    0.12 5.52  142.33  149.55  156.37  2064    1
## y_rep[241]  149.37    0.12 5.54  142.26  149.47  156.54  2067    1
## y_rep[242]  160.35    0.13 5.54  153.33  160.38  167.48  1804    1
## y_rep[243]  160.30    0.13 5.68  153.07  160.21  167.63  1991    1
## y_rep[244]  149.47    0.13 5.61  142.34  149.46  156.72  1897    1
## y_rep[245]  160.29    0.13 5.49  153.24  160.34  167.31  1886    1
## y_rep[246]  149.43    0.13 5.50  142.28  149.38  156.75  1811    1
## y_rep[247]  160.31    0.13 5.57  153.15  160.25  167.48  1893    1
## y_rep[248]  149.63    0.13 5.58  142.70  149.62  156.48  1881    1
## y_rep[249]  160.32    0.12 5.52  153.32  160.40  167.34  2124    1
## y_rep[250]  160.45    0.13 5.58  153.29  160.40  167.62  1894    1
## y_rep[251]  160.31    0.12 5.60  153.07  160.32  167.61  2086    1
## y_rep[252]  160.31    0.13 5.40  153.37  160.23  166.85  1863    1
## y_rep[253]  149.57    0.13 5.65  142.22  149.50  156.82  1919    1
## y_rep[254]  149.55    0.13 5.54  142.52  149.56  156.46  1903    1
## y_rep[255]  149.46    0.12 5.47  142.45  149.31  156.34  1968    1
## y_rep[256]  149.64    0.13 5.56  142.75  149.51  156.80  1802    1
## y_rep[257]  160.17    0.12 5.44  153.17  160.17  167.22  1970    1
## y_rep[258]  149.68    0.12 5.56  142.52  149.60  156.71  2073    1
## y_rep[259]  160.37    0.12 5.55  153.03  160.45  167.69  2011    1
## y_rep[260]  160.32    0.13 5.43  153.48  160.45  167.20  1882    1
## y_rep[261]  160.44    0.13 5.65  153.02  160.52  167.80  1977    1
## y_rep[262]  149.58    0.12 5.49  142.58  149.53  156.77  2001    1
## y_rep[263]  149.56    0.14 5.56  142.32  149.73  156.40  1646    1
## y_rep[264]  149.52    0.13 5.64  142.21  149.60  156.64  1860    1
## y_rep[265]  160.45    0.13 5.57  153.37  160.44  167.49  1908    1
## y_rep[266]  149.55    0.12 5.57  142.31  149.54  156.57  2128    1
## y_rep[267]  160.55    0.13 5.64  153.17  160.61  167.77  1921    1
## y_rep[268]  149.57    0.12 5.49  142.72  149.57  156.62  2050    1
## y_rep[269]  149.62    0.14 5.73  142.34  149.51  156.88  1763    1
## y_rep[270]  160.34    0.13 5.52  153.31  160.16  167.50  1920    1
## y_rep[271]  160.36    0.13 5.55  153.19  160.40  167.27  1828    1
## y_rep[272]  149.45    0.13 5.61  142.31  149.35  156.77  1909    1
## y_rep[273]  160.18    0.12 5.56  153.26  160.33  167.25  2034    1
## y_rep[274]  160.43    0.12 5.55  153.17  160.51  167.35  1998    1
## y_rep[275]  149.83    0.12 5.59  142.77  149.74  156.90  2161    1
## y_rep[276]  160.38    0.12 5.47  153.27  160.49  167.20  2199    1
## y_rep[277]  149.68    0.13 5.52  142.69  149.76  156.77  1858    1
## y_rep[278]  149.29    0.13 5.55  142.20  149.43  156.31  1801    1
## y_rep[279]  160.07    0.12 5.45  153.07  160.08  166.99  1958    1
## y_rep[280]  149.51    0.13 5.52  142.48  149.44  156.54  1897    1
## y_rep[281]  149.49    0.12 5.53  142.22  149.52  156.55  2072    1
## y_rep[282]  160.56    0.13 5.76  153.26  160.56  168.04  1916    1
## y_rep[283]  149.73    0.13 5.49  142.63  149.84  156.76  1877    1
## y_rep[284]  160.16    0.13 5.48  153.18  160.02  167.19  1702    1
## y_rep[285]  149.76    0.13 5.43  142.75  149.79  156.72  1684    1
## y_rep[286]  160.24    0.13 5.47  153.25  160.48  167.06  1898    1
## y_rep[287]  160.32    0.12 5.63  153.22  160.36  167.55  2133    1
## y_rep[288]  160.26    0.13 5.61  153.17  160.36  167.15  1932    1
## y_rep[289]  149.39    0.12 5.41  142.53  149.30  156.58  1923    1
## y_rep[290]  160.23    0.13 5.50  153.12  160.15  167.17  1850    1
## y_rep[291]  160.26    0.11 5.46  153.21  160.35  167.24  2354    1
## y_rep[292]  160.43    0.12 5.59  153.22  160.42  167.54  2014    1
## y_rep[293]  160.11    0.12 5.55  152.91  160.05  167.22  2217    1
## y_rep[294]  149.30    0.13 5.66  142.13  149.34  156.49  1853    1
## y_rep[295]  160.28    0.12 5.60  153.31  160.21  167.50  2137    1
## y_rep[296]  149.64    0.12 5.52  142.54  149.53  156.85  2123    1
## y_rep[297]  149.51    0.12 5.65  142.42  149.49  156.75  2058    1
## y_rep[298]  149.45    0.12 5.39  142.58  149.50  156.19  1922    1
## y_rep[299]  160.47    0.12 5.39  153.55  160.51  167.65  1958    1
## y_rep[300]  160.41    0.13 5.58  153.38  160.50  167.84  1740    1
## y_rep[301]  149.82    0.13 5.47  142.74  149.83  156.87  1894    1
## y_rep[302]  160.24    0.12 5.56  153.26  160.13  167.07  2182    1
## y_rep[303]  149.55    0.13 5.45  142.61  149.57  156.55  1851    1
## y_rep[304]  160.29    0.12 5.45  153.15  160.21  167.45  2010    1
## y_rep[305]  160.21    0.13 5.52  153.27  160.16  167.29  1839    1
## y_rep[306]  160.32    0.13 5.65  153.18  160.33  167.41  1844    1
## y_rep[307]  160.20    0.13 5.60  152.97  160.07  167.37  1818    1
## y_rep[308]  149.65    0.13 5.64  142.54  149.64  156.78  1946    1
## y_rep[309]  149.40    0.12 5.57  142.64  149.32  156.52  2113    1
## y_rep[310]  149.55    0.13 5.55  142.37  149.71  156.27  1957    1
## y_rep[311]  149.41    0.12 5.60  142.45  149.35  156.54  2020    1
## y_rep[312]  149.46    0.13 5.44  142.40  149.53  156.34  1735    1
## y_rep[313]  160.40    0.12 5.63  153.30  160.41  167.79  2254    1
## y_rep[314]  149.60    0.13 5.53  142.61  149.69  156.48  1899    1
## y_rep[315]  160.52    0.13 5.63  153.17  160.56  167.96  2024    1
## y_rep[316]  149.59    0.13 5.64  142.56  149.63  156.76  2003    1
## y_rep[317]  160.40    0.12 5.52  153.35  160.45  167.52  2013    1
## y_rep[318]  149.61    0.12 5.58  142.60  149.51  156.79  2008    1
## y_rep[319]  160.32    0.12 5.64  153.10  160.21  167.60  2178    1
## y_rep[320]  149.61    0.13 5.54  142.50  149.74  156.49  1959    1
## y_rep[321]  160.25    0.14 5.62  153.08  160.29  167.35  1508    1
## y_rep[322]  149.35    0.12 5.54  142.38  149.41  156.50  2120    1
## y_rep[323]  160.34    0.12 5.45  152.85  160.42  167.21  2009    1
## y_rep[324]  160.37    0.12 5.49  153.38  160.39  167.46  1998    1
## y_rep[325]  149.54    0.13 5.67  142.30  149.45  156.93  1950    1
## y_rep[326]  149.47    0.12 5.52  142.38  149.44  156.36  2017    1
## y_rep[327]  149.69    0.12 5.47  142.78  149.73  156.69  1958    1
## y_rep[328]  160.40    0.13 5.63  153.06  160.48  167.50  1846    1
## y_rep[329]  160.36    0.13 5.70  153.02  160.47  167.55  1927    1
## y_rep[330]  160.25    0.14 5.68  153.12  160.30  167.67  1748    1
## y_rep[331]  160.17    0.12 5.52  152.98  160.16  166.99  2115    1
## y_rep[332]  159.95    0.13 5.49  152.56  159.98  166.96  1878    1
## y_rep[333]  149.59    0.12 5.52  142.73  149.51  156.59  1995    1
## y_rep[334]  160.47    0.12 5.54  153.34  160.58  167.28  2027    1
## y_rep[335]  149.45    0.13 5.75  141.82  149.49  156.96  1962    1
## y_rep[336]  160.15    0.12 5.49  153.06  160.19  167.13  2158    1
## y_rep[337]  149.50    0.13 5.64  142.01  149.63  156.61  1944    1
## y_rep[338]  149.55    0.12 5.60  142.19  149.50  156.61  2034    1
## y_rep[339]  149.53    0.12 5.69  142.30  149.49  156.83  2216    1
## y_rep[340]  149.38    0.13 5.70  142.14  149.38  156.77  1841    1
## y_rep[341]  149.58    0.12 5.49  142.73  149.64  156.64  2087    1
## y_rep[342]  160.07    0.13 5.62  152.81  160.10  167.15  1908    1
## y_rep[343]  149.35    0.13 5.51  142.24  149.37  156.32  1792    1
## y_rep[344]  160.18    0.12 5.66  152.80  160.15  167.24  2055    1
## y_rep[345]  149.43    0.14 5.70  142.28  149.35  156.56  1765    1
## y_rep[346]  160.48    0.13 5.67  153.31  160.50  167.72  1953    1
## y_rep[347]  149.67    0.13 5.53  142.60  149.71  156.75  1760    1
## y_rep[348]  149.65    0.13 5.56  142.51  149.82  156.58  1889    1
## y_rep[349]  149.43    0.12 5.56  142.53  149.33  156.46  2048    1
## y_rep[350]  160.38    0.12 5.50  153.43  160.34  167.51  1992    1
## y_rep[351]  149.44    0.13 5.50  142.36  149.40  156.42  1848    1
## y_rep[352]  160.34    0.13 5.49  153.33  160.26  167.39  1849    1
## lp__       -781.30    0.04 1.19 -782.90 -780.98 -780.12   743    1
## 
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:54:46 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

4.2 Index variable

Here we consider the variable male as an index variable, with value 1 whenever the person is a female and a value 2 whenever the person is a male.

howell2$male.index <- ifelse(howell2$male==1,2,1)
m5.2.data <- list(N = nrow(howell2), height = howell2$height, male = as.integer(howell2$male.index), n = length(unique(howell2$male.index)))
str(m5.2.data)
## List of 4
##  $ N     : int 352
##  $ height: num [1:352] 152 140 137 157 145 ...
##  $ male  : int [1:352] 2 1 1 2 1 2 1 2 1 2 ...
##  $ n     : int 2

Model equation:

\[\begin{align*} h_{i} &\sim \mathcal{N}(\mu_{i} \, , \, \sigma^2) \\[4pt] \mu_{i} &= \alpha_{\text{SEX}[\textit{i}]}\\[4pt] \alpha_{j} &\sim \mathcal{N}(178,20) \quad \text{, for} \, j = 1..2\\[4pt] \sigma^2 &\sim \mathcal{U}(0,5)\\[4pt] \end{align*}\]

data {                                                     
  int<lower=0> N;                                          // number of observations
  int<lower=0> n;                                          // number of levels of the categorical variable
  vector[N] height;                                        // response variables
  int<lower=1,upper=n> male[N];                            // categorical variable
}

parameters {                                          
  vector[n] beta;    // real beta[n];  (idem)               // intercepts of the categorical variable
  real sigma;
}

transformed parameters {
vector[N] mu;                                              // create one variable with predictions for each obs
for (i in 1:N)                                             // loop for all cases 
mu[i] = beta[male[i]];  
}

model {
  height ~ normal(mu ,sigma);                              // likelihood
  sigma ~ cauchy(0,5);
}

generated quantities {
  vector[N] y_rep;

 for(i in 1:N)
 y_rep[i] = normal_rng(mu[i], sigma);
}
fit.m5_2 <- sampling(m5.2, data = m5.2.data, iter = 1000, chains = 4, cores = 4)
print(fit.m5_2, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: 0da8a26b283bfab1846d7d15060893cd.
## 4 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=2000.
## 
##               mean se_mean   sd     10%     50%     90% n_eff Rhat
## beta[1]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## beta[2]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## sigma         5.55    0.00 0.20    5.29    5.54    5.81  1800    1
## mu[1]       160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[2]       149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[3]       149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[4]       160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[5]       149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[6]       160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[7]       149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[8]       160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[9]       149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[10]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[11]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[12]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[13]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[14]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[15]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[16]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[17]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[18]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[19]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[20]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[21]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[22]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[23]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[24]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[25]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[26]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[27]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[28]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[29]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[30]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[31]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[32]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[33]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[34]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[35]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[36]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[37]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[38]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[39]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[40]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[41]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[42]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[43]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[44]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[45]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[46]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[47]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[48]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[49]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[50]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[51]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[52]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[53]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[54]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[55]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[56]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[57]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[58]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[59]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[60]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[61]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[62]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[63]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[64]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[65]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[66]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[67]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[68]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[69]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[70]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[71]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[72]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[73]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[74]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[75]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[76]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[77]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[78]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[79]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[80]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[81]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[82]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[83]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[84]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[85]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[86]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[87]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[88]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[89]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[90]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[91]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[92]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[93]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[94]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[95]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[96]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[97]      149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[98]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[99]      160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[100]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[101]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[102]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[103]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[104]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[105]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[106]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[107]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[108]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[109]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[110]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[111]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[112]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[113]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[114]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[115]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[116]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[117]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[118]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[119]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[120]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[121]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[122]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[123]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[124]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[125]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[126]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[127]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[128]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[129]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[130]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[131]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[132]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[133]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[134]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[135]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[136]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[137]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[138]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[139]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[140]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[141]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[142]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[143]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[144]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[145]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[146]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[147]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[148]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[149]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[150]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[151]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[152]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[153]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[154]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[155]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[156]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[157]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[158]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[159]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[160]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[161]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[162]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[163]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[164]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[165]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[166]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[167]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[168]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[169]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[170]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[171]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[172]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[173]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[174]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[175]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[176]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[177]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[178]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[179]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[180]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[181]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[182]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[183]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[184]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[185]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[186]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[187]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[188]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[189]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[190]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[191]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[192]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[193]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[194]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[195]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[196]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[197]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[198]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[199]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[200]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[201]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[202]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[203]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[204]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[205]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[206]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[207]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[208]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[209]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[210]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[211]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[212]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[213]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[214]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[215]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[216]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[217]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[218]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[219]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[220]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[221]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[222]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[223]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[224]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[225]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[226]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[227]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[228]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[229]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[230]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[231]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[232]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[233]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[234]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[235]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[236]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[237]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[238]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[239]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[240]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[241]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[242]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[243]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[244]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[245]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[246]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[247]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[248]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[249]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[250]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[251]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[252]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[253]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[254]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[255]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[256]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[257]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[258]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[259]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[260]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[261]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[262]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[263]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[264]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[265]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[266]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[267]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[268]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[269]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[270]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[271]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[272]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[273]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[274]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[275]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[276]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[277]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[278]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[279]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[280]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[281]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[282]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[283]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[284]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[285]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[286]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[287]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[288]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[289]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[290]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[291]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[292]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[293]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[294]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[295]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[296]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[297]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[298]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[299]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[300]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[301]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[302]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[303]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[304]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[305]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[306]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[307]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[308]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[309]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[310]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[311]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[312]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[313]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[314]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[315]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[316]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[317]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[318]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[319]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[320]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[321]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[322]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[323]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[324]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[325]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[326]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[327]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[328]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[329]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[330]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[331]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[332]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[333]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[334]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[335]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[336]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[337]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[338]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[339]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[340]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[341]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[342]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[343]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[344]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[345]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[346]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[347]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[348]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[349]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[350]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## mu[351]     149.51    0.01 0.40  148.99  149.51  150.01  1825    1
## mu[352]     160.36    0.01 0.46  159.77  160.37  160.93  1728    1
## y_rep[1]    160.40    0.12 5.61  153.41  160.36  167.66  2095    1
## y_rep[2]    149.15    0.13 5.57  142.02  149.10  156.26  1974    1
## y_rep[3]    149.55    0.12 5.47  142.54  149.56  156.49  2057    1
## y_rep[4]    160.35    0.13 5.60  153.11  160.43  167.58  1911    1
## y_rep[5]    149.36    0.12 5.50  142.32  149.32  156.38  2074    1
## y_rep[6]    160.35    0.12 5.57  153.14  160.30  167.29  2285    1
## y_rep[7]    149.65    0.13 5.61  142.28  149.63  156.83  1978    1
## y_rep[8]    160.25    0.13 5.61  153.07  160.30  167.16  1974    1
## y_rep[9]    149.61    0.13 5.71  142.30  149.52  157.01  1992    1
## y_rep[10]   160.40    0.12 5.60  153.24  160.27  167.82  2093    1
## y_rep[11]   149.73    0.12 5.42  142.86  149.73  156.75  1962    1
## y_rep[12]   160.36    0.12 5.52  153.38  160.33  167.46  2019    1
## y_rep[13]   149.46    0.12 5.62  142.39  149.39  156.63  2055    1
## y_rep[14]   149.59    0.14 5.69  142.44  149.45  156.67  1764    1
## y_rep[15]   149.61    0.13 5.55  142.43  149.53  156.46  1926    1
## y_rep[16]   160.36    0.13 5.52  153.35  160.37  167.52  1916    1
## y_rep[17]   160.49    0.13 5.59  153.56  160.38  167.66  1902    1
## y_rep[18]   149.65    0.13 5.56  142.41  149.58  156.70  1957    1
## y_rep[19]   160.49    0.13 5.64  153.47  160.44  167.67  1849    1
## y_rep[20]   149.45    0.13 5.63  142.48  149.47  156.57  1946    1
## y_rep[21]   160.43    0.13 5.66  153.49  160.18  167.93  1793    1
## y_rep[22]   149.32    0.13 5.59  142.13  149.45  156.62  1951    1
## y_rep[23]   149.28    0.12 5.50  142.22  149.28  156.15  2055    1
## y_rep[24]   160.50    0.14 5.57  153.31  160.59  167.53  1667    1
## y_rep[25]   160.38    0.13 5.62  153.50  160.33  167.62  1907    1
## y_rep[26]   149.62    0.13 5.63  142.45  149.66  156.79  1967    1
## y_rep[27]   149.49    0.13 5.58  142.45  149.49  156.62  1962    1
## y_rep[28]   149.70    0.12 5.61  142.50  149.69  156.91  2090    1
## y_rep[29]   149.42    0.12 5.42  142.62  149.30  156.55  1908    1
## y_rep[30]   149.47    0.12 5.54  142.18  149.56  156.58  2058    1
## y_rep[31]   149.52    0.12 5.44  142.62  149.49  156.34  1922    1
## y_rep[32]   160.45    0.13 5.65  153.27  160.55  167.82  1885    1
## y_rep[33]   160.32    0.13 5.51  153.31  160.34  167.39  1937    1
## y_rep[34]   160.29    0.13 5.57  153.30  160.33  167.31  1908    1
## y_rep[35]   160.40    0.13 5.68  153.17  160.61  167.71  1929    1
## y_rep[36]   149.68    0.12 5.44  142.67  149.62  156.79  2107    1
## y_rep[37]   149.66    0.13 5.47  142.78  149.71  156.73  1835    1
## y_rep[38]   160.45    0.12 5.57  153.21  160.19  167.72  2015    1
## y_rep[39]   149.36    0.13 5.67  142.17  149.27  156.70  1975    1
## y_rep[40]   160.49    0.14 5.81  153.14  160.32  167.90  1816    1
## y_rep[41]   160.46    0.12 5.49  153.27  160.48  167.50  1947    1
## y_rep[42]   160.45    0.12 5.54  153.34  160.52  167.78  2097    1
## y_rep[43]   149.37    0.12 5.64  142.35  149.34  156.75  2045    1
## y_rep[44]   149.31    0.13 5.66  141.97  149.18  156.46  1928    1
## y_rep[45]   149.33    0.13 5.58  142.12  149.47  156.41  1955    1
## y_rep[46]   149.53    0.13 5.59  142.25  149.60  156.69  1767    1
## y_rep[47]   160.33    0.13 5.66  153.00  160.49  167.50  1911    1
## y_rep[48]   149.56    0.13 5.60  142.27  149.76  156.57  1963    1
## y_rep[49]   160.25    0.12 5.63  152.99  160.22  167.39  2110    1
## y_rep[50]   149.39    0.13 5.67  142.10  149.34  156.61  1929    1
## y_rep[51]   149.37    0.14 5.76  142.07  149.36  156.81  1646    1
## y_rep[52]   149.38    0.13 5.64  142.18  149.38  156.74  1909    1
## y_rep[53]   160.19    0.12 5.45  153.03  160.22  167.35  2037    1
## y_rep[54]   160.21    0.13 5.61  153.09  160.35  167.42  1778    1
## y_rep[55]   149.68    0.13 5.57  142.59  149.84  156.67  1938    1
## y_rep[56]   160.32    0.12 5.43  153.22  160.32  167.23  2032    1
## y_rep[57]   149.52    0.12 5.55  142.26  149.47  156.59  2088    1
## y_rep[58]   160.43    0.13 5.51  153.27  160.52  167.74  1840    1
## y_rep[59]   149.44    0.13 5.55  142.35  149.63  156.54  1889    1
## y_rep[60]   160.31    0.12 5.45  153.57  160.06  167.44  1984    1
## y_rep[61]   149.64    0.13 5.56  142.33  149.73  156.76  1937    1
## y_rep[62]   149.59    0.14 5.72  142.52  149.55  156.98  1727    1
## y_rep[63]   160.50    0.12 5.48  153.57  160.25  167.55  1945    1
## y_rep[64]   160.51    0.13 5.47  153.66  160.47  167.61  1816    1
## y_rep[65]   149.49    0.13 5.56  142.44  149.45  156.53  1839    1
## y_rep[66]   160.36    0.13 5.47  153.23  160.39  167.34  1827    1
## y_rep[67]   149.64    0.13 5.58  142.62  149.58  156.72  1853    1
## y_rep[68]   149.24    0.12 5.60  141.90  149.11  156.35  2082    1
## y_rep[69]   160.22    0.12 5.49  153.46  160.19  167.43  1966    1
## y_rep[70]   149.46    0.13 5.50  142.63  149.41  156.80  1747    1
## y_rep[71]   160.25    0.12 5.51  152.97  160.46  167.23  2229    1
## y_rep[72]   160.24    0.13 5.49  153.18  160.38  167.11  1866    1
## y_rep[73]   160.45    0.13 5.65  153.06  160.43  167.94  1976    1
## y_rep[74]   149.50    0.12 5.54  142.26  149.57  156.73  2027    1
## y_rep[75]   149.49    0.12 5.54  142.55  149.45  156.61  2032    1
## y_rep[76]   149.62    0.13 5.60  142.39  149.63  156.67  1753    1
## y_rep[77]   149.51    0.13 5.61  142.40  149.68  156.84  1939    1
## y_rep[78]   160.09    0.12 5.50  153.06  160.15  167.12  1970    1
## y_rep[79]   149.61    0.12 5.51  142.47  149.79  156.73  1991    1
## y_rep[80]   149.59    0.13 5.51  142.45  149.44  156.69  1885    1
## y_rep[81]   160.41    0.13 5.48  153.49  160.44  167.32  1919    1
## y_rep[82]   160.27    0.12 5.54  153.08  160.15  167.60  2113    1
## y_rep[83]   149.49    0.12 5.53  142.55  149.54  156.78  2044    1
## y_rep[84]   160.09    0.12 5.46  152.99  160.07  167.21  1979    1
## y_rep[85]   149.46    0.13 5.70  142.39  149.41  156.81  1907    1
## y_rep[86]   160.31    0.12 5.62  152.99  160.34  167.48  2089    1
## y_rep[87]   149.38    0.12 5.53  142.19  149.39  156.52  1977    1
## y_rep[88]   160.32    0.12 5.64  153.27  160.34  167.49  2201    1
## y_rep[89]   149.70    0.12 5.54  142.77  149.63  156.72  2042    1
## y_rep[90]   160.41    0.12 5.49  153.36  160.45  167.60  2057    1
## y_rep[91]   160.27    0.12 5.58  153.14  160.31  167.51  2033    1
## y_rep[92]   160.56    0.13 5.61  153.55  160.47  167.73  1928    1
## y_rep[93]   149.57    0.12 5.46  142.75  149.50  156.56  1974    1
## y_rep[94]   149.41    0.13 5.41  142.47  149.35  156.46  1761    1
## y_rep[95]   160.46    0.12 5.61  153.22  160.50  167.75  2070    1
## y_rep[96]   160.44    0.13 5.60  153.28  160.57  167.80  1843    1
## y_rep[97]   149.46    0.12 5.58  142.32  149.52  156.70  2054    1
## y_rep[98]   160.17    0.12 5.64  153.05  160.17  167.54  2175    1
## y_rep[99]   160.38    0.12 5.47  153.49  160.36  167.48  2025    1
## y_rep[100]  149.44    0.13 5.59  142.17  149.59  156.64  1936    1
## y_rep[101]  160.46    0.12 5.45  153.57  160.42  167.68  2118    1
## y_rep[102]  149.52    0.13 5.56  142.57  149.55  156.73  1824    1
## y_rep[103]  149.61    0.13 5.66  142.20  149.72  157.00  1917    1
## y_rep[104]  160.46    0.14 5.68  153.04  160.48  167.74  1688    1
## y_rep[105]  160.39    0.13 5.54  153.48  160.44  167.46  1823    1
## y_rep[106]  149.62    0.13 5.70  142.29  149.64  156.90  2069    1
## y_rep[107]  149.54    0.12 5.63  142.53  149.60  156.85  2090    1
## y_rep[108]  149.63    0.13 5.57  142.47  149.55  156.70  1982    1
## y_rep[109]  160.35    0.12 5.63  153.16  160.21  167.77  2170    1
## y_rep[110]  149.43    0.13 5.73  142.08  149.41  156.69  2051    1
## y_rep[111]  160.42    0.12 5.69  153.25  160.30  167.73  2104    1
## y_rep[112]  149.54    0.12 5.50  142.41  149.60  156.77  1948    1
## y_rep[113]  149.48    0.12 5.49  142.45  149.55  156.64  1990    1
## y_rep[114]  160.16    0.13 5.58  152.97  160.24  167.29  1891    1
## y_rep[115]  160.52    0.12 5.46  153.47  160.38  167.45  2150    1
## y_rep[116]  149.51    0.13 5.60  142.22  149.41  156.80  1738    1
## y_rep[117]  160.31    0.13 5.37  153.49  160.24  167.27  1831    1
## y_rep[118]  160.02    0.13 5.47  152.93  160.12  167.13  1884    1
## y_rep[119]  149.54    0.13 5.51  142.31  149.51  156.55  1867    1
## y_rep[120]  149.60    0.12 5.54  142.56  149.60  156.66  2091    1
## y_rep[121]  160.55    0.12 5.57  153.38  160.51  167.83  2186    1
## y_rep[122]  149.38    0.13 5.58  142.01  149.54  156.56  1881    1
## y_rep[123]  160.37    0.12 5.48  153.48  160.36  167.13  2171    1
## y_rep[124]  149.49    0.13 5.39  142.57  149.58  156.33  1801    1
## y_rep[125]  149.65    0.12 5.60  142.54  149.55  157.16  2043    1
## y_rep[126]  149.28    0.13 5.66  142.11  149.15  156.74  1774    1
## y_rep[127]  149.83    0.13 5.78  142.42  149.86  157.26  2043    1
## y_rep[128]  160.14    0.13 5.66  153.10  160.04  167.44  2030    1
## y_rep[129]  149.40    0.12 5.50  142.28  149.51  156.49  2113    1
## y_rep[130]  149.47    0.12 5.57  142.56  149.35  156.85  2015    1
## y_rep[131]  160.38    0.14 5.57  153.38  160.29  167.44  1626    1
## y_rep[132]  160.37    0.13 5.47  153.27  160.51  167.42  1886    1
## y_rep[133]  149.50    0.13 5.59  142.52  149.38  156.92  1840    1
## y_rep[134]  160.38    0.12 5.51  153.33  160.43  167.58  2113    1
## y_rep[135]  149.31    0.12 5.54  142.19  149.38  156.16  2041    1
## y_rep[136]  160.27    0.12 5.53  153.23  160.13  167.57  2122    1
## y_rep[137]  160.18    0.12 5.57  152.97  160.09  167.19  2222    1
## y_rep[138]  160.28    0.12 5.55  153.26  160.23  167.46  2121    1
## y_rep[139]  149.38    0.13 5.48  142.38  149.25  156.32  1809    1
## y_rep[140]  160.11    0.12 5.52  153.28  160.15  167.29  1989    1
## y_rep[141]  149.63    0.12 5.49  142.68  149.56  156.59  2070    1
## y_rep[142]  149.47    0.13 5.58  142.33  149.51  156.55  1934    1
## y_rep[143]  149.44    0.12 5.40  142.50  149.47  156.36  1946    1
## y_rep[144]  149.49    0.12 5.48  142.38  149.49  156.40  2042    1
## y_rep[145]  149.48    0.13 5.45  142.76  149.40  156.48  1809    1
## y_rep[146]  160.36    0.13 5.57  153.47  160.36  167.40  1842    1
## y_rep[147]  149.69    0.12 5.53  142.58  149.76  156.79  1963    1
## y_rep[148]  149.43    0.14 5.70  142.16  149.46  156.66  1770    1
## y_rep[149]  160.14    0.13 5.59  152.72  160.17  167.25  1819    1
## y_rep[150]  149.26    0.12 5.47  142.21  149.18  156.37  2141    1
## y_rep[151]  160.36    0.13 5.79  152.84  160.44  167.81  1930    1
## y_rep[152]  149.52    0.13 5.66  142.40  149.43  156.81  1849    1
## y_rep[153]  160.39    0.12 5.49  153.51  160.33  167.52  2018    1
## y_rep[154]  149.45    0.13 5.61  142.33  149.50  156.48  1938    1
## y_rep[155]  160.21    0.12 5.58  153.17  160.06  167.49  2083    1
## y_rep[156]  149.29    0.13 5.57  142.02  149.40  156.39  1836    1
## y_rep[157]  149.69    0.12 5.43  142.79  149.74  156.86  1891    1
## y_rep[158]  160.22    0.13 5.71  153.16  160.29  167.55  1950    1
## y_rep[159]  160.46    0.12 5.57  153.19  160.47  167.59  2008    1
## y_rep[160]  149.57    0.13 5.49  142.67  149.45  156.76  1876    1
## y_rep[161]  160.26    0.13 5.67  153.12  160.25  167.47  1996    1
## y_rep[162]  149.64    0.12 5.55  142.75  149.70  156.69  2043    1
## y_rep[163]  160.27    0.13 5.68  152.83  160.16  167.54  1850    1
## y_rep[164]  160.41    0.14 5.76  153.29  160.63  167.59  1788    1
## y_rep[165]  160.42    0.12 5.68  152.95  160.44  167.77  2098    1
## y_rep[166]  149.25    0.13 5.43  142.32  149.27  156.29  1720    1
## y_rep[167]  160.26    0.12 5.53  153.13  160.26  167.29  1973    1
## y_rep[168]  149.64    0.14 5.64  142.43  149.67  156.85  1722    1
## y_rep[169]  149.28    0.13 5.61  142.20  149.34  156.40  1974    1
## y_rep[170]  149.48    0.13 5.64  142.42  149.42  156.85  1885    1
## y_rep[171]  149.31    0.13 5.48  142.25  149.18  156.56  1672    1
## y_rep[172]  160.43    0.12 5.42  153.39  160.50  167.28  1984    1
## y_rep[173]  149.25    0.13 5.69  142.02  149.34  156.64  1967    1
## y_rep[174]  160.60    0.12 5.60  153.64  160.56  167.92  2053    1
## y_rep[175]  149.67    0.13 5.62  142.67  149.72  156.80  1870    1
## y_rep[176]  160.13    0.12 5.46  153.14  160.23  166.87  2040    1
## y_rep[177]  149.63    0.14 5.55  142.50  149.71  156.45  1634    1
## y_rep[178]  149.53    0.12 5.67  142.37  149.42  156.81  2172    1
## y_rep[179]  160.21    0.13 5.62  152.87  160.28  167.38  2012    1
## y_rep[180]  149.30    0.12 5.36  142.46  149.31  156.12  2041    1
## y_rep[181]  160.22    0.12 5.54  153.34  160.24  167.20  2034    1
## y_rep[182]  149.34    0.12 5.39  142.23  149.41  156.19  2009    1
## y_rep[183]  149.51    0.13 5.67  142.20  149.47  156.95  2029    1
## y_rep[184]  149.56    0.12 5.43  142.57  149.66  156.34  1930    1
## y_rep[185]  160.37    0.13 5.55  153.20  160.56  167.36  1919    1
## y_rep[186]  160.47    0.12 5.67  153.07  160.47  167.55  2109    1
## y_rep[187]  149.49    0.13 5.62  142.29  149.57  156.70  1776    1
## y_rep[188]  160.42    0.12 5.47  153.30  160.51  167.35  1963    1
## y_rep[189]  149.50    0.13 5.60  142.34  149.39  156.79  1939    1
## y_rep[190]  160.39    0.12 5.63  153.16  160.41  167.40  2075    1
## y_rep[191]  149.55    0.12 5.49  142.62  149.72  156.53  2051    1
## y_rep[192]  160.50    0.13 5.53  153.40  160.51  167.64  1897    1
## y_rep[193]  149.46    0.13 5.68  142.24  149.41  156.99  1969    1
## y_rep[194]  160.33    0.13 5.69  152.90  160.32  167.52  1992    1
## y_rep[195]  149.61    0.13 5.65  142.44  149.65  156.69  1898    1
## y_rep[196]  149.51    0.13 5.62  142.12  149.61  156.48  1793    1
## y_rep[197]  149.51    0.12 5.48  142.35  149.53  156.38  2059    1
## y_rep[198]  160.50    0.13 5.67  153.23  160.44  167.84  2012    1
## y_rep[199]  160.43    0.13 5.63  153.38  160.24  167.75  1832    1
## y_rep[200]  149.31    0.12 5.66  142.34  149.20  156.60  2075    1
## y_rep[201]  160.27    0.13 5.61  153.11  160.24  167.50  1967    1
## y_rep[202]  149.46    0.12 5.63  142.43  149.52  156.46  2146    1
## y_rep[203]  160.42    0.12 5.60  153.34  160.34  167.69  2214    1
## y_rep[204]  149.49    0.13 5.70  142.16  149.39  156.85  1951    1
## y_rep[205]  160.28    0.12 5.41  153.24  160.27  167.09  2019    1
## y_rep[206]  149.64    0.13 5.47  142.83  149.56  156.79  1895    1
## y_rep[207]  149.59    0.12 5.63  142.41  149.48  156.74  2032    1
## y_rep[208]  149.44    0.14 5.51  142.53  149.29  156.62  1609    1
## y_rep[209]  149.59    0.12 5.55  142.65  149.55  156.75  2109    1
## y_rep[210]  149.57    0.13 5.70  142.44  149.55  157.00  1928    1
## y_rep[211]  160.42    0.12 5.55  153.34  160.27  167.59  2065    1
## y_rep[212]  160.50    0.12 5.53  153.56  160.77  167.65  2114    1
## y_rep[213]  160.50    0.12 5.59  153.25  160.48  167.52  2033    1
## y_rep[214]  149.59    0.12 5.67  142.33  149.78  156.63  2066    1
## y_rep[215]  160.19    0.12 5.54  153.32  160.05  167.32  2037    1
## y_rep[216]  160.32    0.12 5.52  153.39  160.35  167.22  1962    1
## y_rep[217]  160.38    0.13 5.61  153.13  160.38  167.73  1972    1
## y_rep[218]  149.49    0.13 5.60  142.30  149.54  156.77  1970    1
## y_rep[219]  160.36    0.12 5.59  153.18  160.39  167.57  2285    1
## y_rep[220]  149.69    0.12 5.67  142.55  149.75  156.65  2061    1
## y_rep[221]  149.44    0.14 5.76  141.92  149.31  157.05  1814    1
## y_rep[222]  149.46    0.13 5.56  142.24  149.37  156.44  1947    1
## y_rep[223]  160.44    0.13 5.60  153.35  160.47  167.40  1996    1
## y_rep[224]  149.26    0.13 5.69  141.87  149.21  156.72  1976    1
## y_rep[225]  160.37    0.12 5.54  153.38  160.28  167.60  2015    1
## y_rep[226]  149.45    0.12 5.65  142.50  149.32  156.65  2092    1
## y_rep[227]  149.56    0.13 5.46  142.81  149.59  156.47  1691    1
## y_rep[228]  160.37    0.13 5.54  153.31  160.28  167.61  1916    1
## y_rep[229]  149.55    0.13 5.74  142.37  149.53  156.91  1994    1
## y_rep[230]  160.39    0.13 5.55  153.13  160.39  167.53  1770    1
## y_rep[231]  160.30    0.12 5.55  153.16  160.29  167.42  1978    1
## y_rep[232]  149.57    0.13 5.57  142.41  149.58  156.73  1833    1
## y_rep[233]  149.75    0.13 5.69  142.52  149.81  157.04  1848    1
## y_rep[234]  160.41    0.12 5.50  153.66  160.32  167.64  2041    1
## y_rep[235]  160.36    0.13 5.51  153.33  160.39  167.28  1903    1
## y_rep[236]  149.57    0.12 5.45  142.78  149.69  156.34  1956    1
## y_rep[237]  160.24    0.14 5.71  152.84  160.26  167.59  1742    1
## y_rep[238]  149.75    0.12 5.40  142.74  149.61  156.60  2105    1
## y_rep[239]  149.35    0.13 5.54  142.29  149.30  156.35  1837    1
## y_rep[240]  149.38    0.12 5.47  142.35  149.40  156.38  2158    1
## y_rep[241]  149.14    0.12 5.44  141.81  149.29  156.15  2185    1
## y_rep[242]  160.35    0.12 5.57  153.12  160.54  167.37  2019    1
## y_rep[243]  160.36    0.13 5.71  152.91  160.34  167.61  2041    1
## y_rep[244]  149.64    0.13 5.59  142.56  149.56  156.82  1992    1
## y_rep[245]  160.60    0.12 5.44  153.67  160.59  167.70  1978    1
## y_rep[246]  149.53    0.13 5.57  142.62  149.35  156.61  1944    1
## y_rep[247]  160.30    0.13 5.67  153.04  160.19  167.72  1978    1
## y_rep[248]  149.57    0.13 5.61  142.34  149.72  156.88  1962    1
## y_rep[249]  160.36    0.13 5.61  153.27  160.35  167.42  1961    1
## y_rep[250]  160.59    0.13 5.58  153.51  160.61  167.53  1913    1
## y_rep[251]  160.39    0.13 5.66  153.02  160.29  167.62  1880    1
## y_rep[252]  160.62    0.12 5.53  153.60  160.61  167.56  2041    1
## y_rep[253]  149.54    0.12 5.61  142.51  149.27  156.89  2051    1
## y_rep[254]  149.59    0.12 5.50  142.38  149.60  156.56  2069    1
## y_rep[255]  149.26    0.12 5.47  142.58  149.04  156.38  1980    1
## y_rep[256]  149.53    0.12 5.58  142.35  149.58  156.67  2155    1
## y_rep[257]  160.22    0.13 5.56  153.13  160.29  167.29  1871    1
## y_rep[258]  149.71    0.13 5.66  142.56  149.70  157.10  1999    1
## y_rep[259]  160.44    0.13 5.62  153.16  160.31  167.61  2001    1
## y_rep[260]  160.34    0.12 5.43  153.34  160.51  167.23  1992    1
## y_rep[261]  160.32    0.12 5.60  153.08  160.36  167.64  2135    1
## y_rep[262]  149.63    0.12 5.53  142.47  149.65  156.52  2039    1
## y_rep[263]  149.55    0.13 5.69  142.21  149.49  156.99  1997    1
## y_rep[264]  149.50    0.14 5.51  142.34  149.64  156.35  1619    1
## y_rep[265]  160.43    0.13 5.55  153.19  160.55  167.27  1895    1
## y_rep[266]  149.59    0.13 5.52  142.30  149.66  156.57  1946    1
## y_rep[267]  160.51    0.14 5.46  153.60  160.74  167.17  1519    1
## y_rep[268]  149.45    0.13 5.56  142.31  149.49  156.41  1843    1
## y_rep[269]  149.40    0.13 5.68  142.02  149.43  156.73  1947    1
## y_rep[270]  160.52    0.13 5.49  153.31  160.60  167.45  1910    1
## y_rep[271]  160.51    0.13 5.72  153.17  160.45  167.86  1951    1
## y_rep[272]  149.74    0.13 5.62  142.62  149.75  157.01  2015    1
## y_rep[273]  160.54    0.13 5.49  153.31  160.64  167.50  1855    1
## y_rep[274]  160.45    0.12 5.57  153.22  160.57  167.62  2010    1
## y_rep[275]  149.51    0.13 5.68  142.23  149.60  156.75  2031    1
## y_rep[276]  160.32    0.13 5.56  153.33  160.41  167.09  1886    1
## y_rep[277]  149.37    0.13 5.63  142.40  149.31  156.61  1987    1
## y_rep[278]  149.49    0.13 5.60  142.11  149.63  156.60  1993    1
## y_rep[279]  160.34    0.13 5.71  153.04  160.19  167.77  1943    1
## y_rep[280]  149.31    0.12 5.53  142.24  149.29  156.70  2034    1
## y_rep[281]  149.42    0.13 5.52  142.26  149.33  156.55  1845    1
## y_rep[282]  160.28    0.13 5.47  153.33  160.22  167.21  1819    1
## y_rep[283]  149.52    0.12 5.54  142.34  149.48  156.67  1992    1
## y_rep[284]  160.22    0.12 5.62  153.15  160.08  167.43  2053    1
## y_rep[285]  149.49    0.12 5.44  142.51  149.29  156.58  2131    1
## y_rep[286]  160.27    0.13 5.70  152.91  160.51  167.45  1984    1
## y_rep[287]  160.32    0.12 5.46  153.31  160.25  167.50  2045    1
## y_rep[288]  160.33    0.13 5.46  153.29  160.37  167.22  1664    1
## y_rep[289]  149.35    0.12 5.52  142.47  149.24  156.61  2090    1
## y_rep[290]  160.31    0.14 5.61  153.35  160.17  167.50  1685    1
## y_rep[291]  160.50    0.14 5.64  153.01  160.64  167.56  1746    1
## y_rep[292]  160.43    0.12 5.52  153.28  160.54  167.38  2053    1
## y_rep[293]  160.42    0.12 5.54  153.39  160.50  167.39  2059    1
## y_rep[294]  149.49    0.12 5.52  142.56  149.46  156.57  2048    1
## y_rep[295]  160.25    0.12 5.52  152.96  160.30  167.36  2022    1
## y_rep[296]  149.51    0.12 5.53  142.64  149.59  156.52  2145    1
## y_rep[297]  149.48    0.12 5.50  142.55  149.33  156.36  2113    1
## y_rep[298]  149.43    0.13 5.69  142.44  149.30  156.64  1927    1
## y_rep[299]  160.32    0.13 5.52  153.29  160.33  167.52  1854    1
## y_rep[300]  160.36    0.12 5.52  153.59  160.36  167.65  2081    1
## y_rep[301]  149.44    0.13 5.77  142.04  149.33  157.09  2056    1
## y_rep[302]  160.32    0.13 5.69  153.02  160.33  167.56  1937    1
## y_rep[303]  149.49    0.13 5.63  142.41  149.48  156.82  1954    1
## y_rep[304]  160.20    0.13 5.61  153.14  160.15  167.30  2000    1
## y_rep[305]  160.26    0.12 5.44  153.16  160.31  166.98  2085    1
## y_rep[306]  160.09    0.12 5.42  152.96  160.11  167.01  2020    1
## y_rep[307]  160.32    0.12 5.54  153.25  160.40  167.13  2161    1
## y_rep[308]  149.28    0.12 5.61  142.24  149.19  156.57  2027    1
## y_rep[309]  149.43    0.13 5.55  142.41  149.30  156.58  1800    1
## y_rep[310]  149.83    0.12 5.55  142.56  149.84  157.05  2078    1
## y_rep[311]  149.53    0.12 5.56  142.42  149.45  156.47  1988    1
## y_rep[312]  149.63    0.13 5.54  142.52  149.81  156.71  1855    1
## y_rep[313]  160.21    0.13 5.75  152.92  160.21  167.53  1867    1
## y_rep[314]  149.35    0.13 5.68  142.20  149.21  156.68  1999    1
## y_rep[315]  160.39    0.13 5.59  153.10  160.53  167.42  1855    1
## y_rep[316]  149.46    0.13 5.60  142.28  149.44  156.72  1955    1
## y_rep[317]  160.16    0.13 5.65  152.94  160.28  167.41  1921    1
## y_rep[318]  149.50    0.12 5.52  142.35  149.57  156.52  2083    1
## y_rep[319]  160.42    0.12 5.49  153.46  160.38  167.48  1971    1
## y_rep[320]  149.31    0.12 5.50  142.36  149.13  156.48  2005    1
## y_rep[321]  160.22    0.13 5.62  152.85  160.26  167.41  1857    1
## y_rep[322]  149.47    0.12 5.52  142.43  149.34  156.51  2091    1
## y_rep[323]  160.29    0.13 5.49  153.15  160.40  167.34  1840    1
## y_rep[324]  160.34    0.12 5.50  153.15  160.48  167.47  2053    1
## y_rep[325]  149.51    0.13 5.59  142.48  149.47  156.81  1948    1
## y_rep[326]  149.52    0.12 5.51  142.55  149.65  156.69  1995    1
## y_rep[327]  149.50    0.13 5.55  142.55  149.49  156.84  1884    1
## y_rep[328]  160.37    0.13 5.63  153.41  160.33  167.69  1961    1
## y_rep[329]  160.45    0.12 5.52  153.62  160.39  167.48  1978    1
## y_rep[330]  160.32    0.13 5.61  152.98  160.34  167.47  1943    1
## y_rep[331]  160.38    0.14 5.44  153.47  160.42  167.39  1473    1
## y_rep[332]  160.31    0.12 5.46  153.34  160.28  167.57  1931    1
## y_rep[333]  149.63    0.12 5.58  142.58  149.62  157.08  2080    1
## y_rep[334]  160.36    0.12 5.56  153.30  160.47  167.45  2068    1
## y_rep[335]  149.22    0.12 5.40  142.23  149.29  156.14  2059    1
## y_rep[336]  160.44    0.13 5.65  153.19  160.58  167.63  1973    1
## y_rep[337]  149.66    0.13 5.55  142.51  149.66  156.78  1949    1
## y_rep[338]  149.67    0.12 5.45  142.57  149.69  156.63  2008    1
## y_rep[339]  149.48    0.13 5.55  142.54  149.44  156.51  1920    1
## y_rep[340]  149.66    0.12 5.55  142.53  149.67  156.77  2011    1
## y_rep[341]  149.60    0.13 5.78  142.29  149.72  157.03  1998    1
## y_rep[342]  160.21    0.13 5.80  152.93  160.10  167.57  1945    1
## y_rep[343]  149.41    0.13 5.62  142.23  149.38  156.62  1806    1
## y_rep[344]  160.48    0.15 5.52  153.63  160.54  167.53  1420    1
## y_rep[345]  149.53    0.12 5.58  142.48  149.56  156.87  2025    1
## y_rep[346]  160.51    0.13 5.54  153.46  160.61  167.40  1795    1
## y_rep[347]  149.42    0.13 5.71  142.19  149.44  156.65  1918    1
## y_rep[348]  149.28    0.12 5.49  142.45  149.21  156.35  2088    1
## y_rep[349]  149.49    0.13 5.54  142.26  149.50  156.54  1859    1
## y_rep[350]  160.26    0.13 5.45  153.43  160.16  167.38  1808    1
## y_rep[351]  149.37    0.12 5.44  142.29  149.27  156.38  1939    1
## y_rep[352]  160.39    0.13 5.71  153.01  160.50  167.76  1953    1
## lp__       -779.74    0.04 1.24 -781.47 -779.44 -778.50  1045    1
## 
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:55:46 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

Corresponds to a frequentist model with the intercept fixed to zero:

summary(lm(height~as.factor(male)-1,data=howell2))
## 
## Call:
## lm(formula = height ~ as.factor(male) - 1, data = howell2)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -20.6585  -3.4635   0.2965   3.4715  18.7115 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## as.factor(male)0 149.5135     0.4049   369.2   <2e-16 ***
## as.factor(male)1 160.3585     0.4311   372.0   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.537 on 350 degrees of freedom
## Multiple R-squared:  0.9987, Adjusted R-squared:  0.9987 
## F-statistic: 1.374e+05 on 2 and 350 DF,  p-value: < 2.2e-16

Even if you use a index variable (with values 1 or 2), you can also use the same notation as for dummy variables

X <- model.matrix(~male.index, howell2)
m5.2.bis.data <- list(N = nrow(howell2), height = howell2$height, X = X, n = length(unique(howell2$male.index)))
str(m5.2.bis.data)
## List of 4
##  $ N     : int 352
##  $ height: num [1:352] 152 140 137 157 145 ...
##  $ X     : num [1:352, 1:2] 1 1 1 1 1 1 1 1 1 1 ...
##   ..- attr(*, "dimnames")=List of 2
##   .. ..$ : chr [1:352] "1" "2" "3" "4" ...
##   .. ..$ : chr [1:2] "(Intercept)" "male.index"
##   ..- attr(*, "assign")= int [1:2] 0 1
##  $ n     : int 2
data {                                                     
  int<lower=0> N;                                          // number of observations
  int<lower=0> n;                                          // number of levels of the categorical variable
  vector[N] height;                                        // response variables
  matrix[N,n] X;                                           // categorical variable
}

parameters {                                          
  vector[n] beta;                                         // intercepts of the categorical variable
  real sigma;
}

transformed parameters {
vector[N] mu;                                             
mu = X*beta;  
}

model {
  height ~ normal(mu ,sigma);                              // likelihood
  sigma ~ cauchy(0,5);
  beta[1] ~ normal(178,20);
  beta[2] ~ normal(0,10);
}

generated quantities {
  vector[N] y_rep;
 for(i in 1:N)
 y_rep[i] = normal_rng(mu[i], sigma);
}
fit.m5_2_bis <- sampling(m5.2.bis, data = m5.2.bis.data, iter = 1000, chains = 4, cores = 4)
print(fit.m5_2_bis, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: ea1df6c828dbad449e9a4c6c6321058c.
## 4 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=2000.
## 
##               mean se_mean   sd     10%     50%     90% n_eff Rhat
## beta[1]     138.80    0.03 0.90  137.61  138.82  139.97   724    1
## beta[2]      10.76    0.02 0.58   10.02   10.76   11.50   736    1
## sigma         5.57    0.01 0.21    5.30    5.55    5.84   942    1
## mu[1]       160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[2]       149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[3]       149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[4]       160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[5]       149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[6]       160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[7]       149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[8]       160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[9]       149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[10]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[11]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[12]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[13]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[14]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[15]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[16]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[17]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[18]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[19]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[20]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[21]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[22]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[23]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[24]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[25]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[26]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[27]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[28]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[29]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[30]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[31]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[32]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[33]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[34]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[35]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[36]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[37]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[38]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[39]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[40]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[41]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[42]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[43]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[44]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[45]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[46]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[47]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[48]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[49]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[50]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[51]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[52]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[53]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[54]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[55]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[56]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[57]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[58]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[59]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[60]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[61]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[62]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[63]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[64]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[65]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[66]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[67]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[68]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[69]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[70]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[71]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[72]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[73]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[74]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[75]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[76]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[77]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[78]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[79]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[80]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[81]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[82]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[83]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[84]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[85]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[86]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[87]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[88]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[89]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[90]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[91]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[92]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[93]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[94]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[95]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[96]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[97]      149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[98]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[99]      160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[100]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[101]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[102]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[103]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[104]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[105]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[106]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[107]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[108]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[109]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[110]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[111]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[112]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[113]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[114]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[115]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[116]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[117]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[118]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[119]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[120]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[121]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[122]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[123]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[124]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[125]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[126]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[127]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[128]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[129]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[130]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[131]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[132]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[133]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[134]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[135]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[136]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[137]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[138]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[139]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[140]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[141]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[142]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[143]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[144]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[145]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[146]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[147]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[148]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[149]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[150]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[151]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[152]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[153]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[154]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[155]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[156]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[157]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[158]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[159]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[160]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[161]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[162]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[163]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[164]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[165]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[166]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[167]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[168]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[169]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[170]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[171]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[172]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[173]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[174]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[175]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[176]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[177]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[178]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[179]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[180]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[181]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[182]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[183]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[184]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[185]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[186]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[187]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[188]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[189]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[190]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[191]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[192]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[193]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[194]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[195]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[196]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[197]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[198]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[199]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[200]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[201]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[202]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[203]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[204]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[205]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[206]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[207]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[208]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[209]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[210]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[211]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[212]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[213]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[214]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[215]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[216]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[217]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[218]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[219]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[220]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[221]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[222]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[223]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[224]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[225]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[226]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[227]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[228]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[229]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[230]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[231]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[232]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[233]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[234]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[235]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[236]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[237]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[238]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[239]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[240]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[241]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[242]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[243]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[244]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[245]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[246]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[247]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[248]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[249]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[250]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[251]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[252]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[253]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[254]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[255]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[256]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[257]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[258]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[259]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[260]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[261]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[262]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[263]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[264]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[265]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[266]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[267]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[268]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[269]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[270]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[271]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[272]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[273]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[274]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[275]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[276]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[277]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[278]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[279]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[280]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[281]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[282]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[283]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[284]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[285]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[286]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[287]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[288]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[289]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[290]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[291]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[292]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[293]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[294]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[295]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[296]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[297]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[298]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[299]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[300]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[301]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[302]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[303]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[304]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[305]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[306]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[307]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[308]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[309]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[310]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[311]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[312]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[313]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[314]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[315]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[316]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[317]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[318]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[319]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[320]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[321]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[322]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[323]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[324]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[325]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[326]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[327]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[328]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[329]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[330]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[331]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[332]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[333]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[334]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[335]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[336]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[337]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[338]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[339]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[340]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[341]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[342]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[343]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[344]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[345]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[346]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[347]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[348]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[349]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[350]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## mu[351]     149.56    0.01 0.40  149.02  149.56  150.08   956    1
## mu[352]     160.32    0.01 0.42  159.78  160.33  160.85  1327    1
## y_rep[1]    160.19    0.13 5.74  152.80  160.16  167.43  1963    1
## y_rep[2]    149.63    0.12 5.66  142.21  149.78  156.81  2058    1
## y_rep[3]    149.57    0.12 5.53  142.31  149.63  156.57  2020    1
## y_rep[4]    160.08    0.12 5.67  152.94  160.10  167.17  2069    1
## y_rep[5]    149.32    0.12 5.53  142.21  149.42  156.54  1963    1
## y_rep[6]    160.38    0.13 5.58  153.43  160.25  167.56  1863    1
## y_rep[7]    149.63    0.13 5.84  142.06  149.57  157.18  1956    1
## y_rep[8]    160.29    0.13 5.71  153.01  160.17  167.75  2012    1
## y_rep[9]    149.42    0.13 5.63  142.22  149.52  156.49  1866    1
## y_rep[10]   160.41    0.13 5.68  153.01  160.67  167.58  1967    1
## y_rep[11]   149.47    0.13 5.65  142.27  149.47  156.60  2018    1
## y_rep[12]   160.29    0.12 5.55  153.25  160.26  167.04  2036    1
## y_rep[13]   149.46    0.13 5.64  142.12  149.47  156.51  1952    1
## y_rep[14]   149.56    0.13 5.68  141.90  149.71  156.75  2025    1
## y_rep[15]   149.55    0.12 5.50  142.45  149.63  156.60  1964    1
## y_rep[16]   160.35    0.13 5.46  153.48  160.16  167.38  1894    1
## y_rep[17]   160.08    0.12 5.52  153.16  160.11  166.99  1959    1
## y_rep[18]   149.66    0.13 5.63  142.44  149.79  156.76  1904    1
## y_rep[19]   160.30    0.13 5.51  153.11  160.36  167.31  1931    1
## y_rep[20]   149.57    0.13 5.71  141.97  149.71  156.93  1885    1
## y_rep[21]   160.14    0.13 5.73  152.91  160.16  167.65  2022    1
## y_rep[22]   149.63    0.13 5.65  142.48  149.59  156.86  1944    1
## y_rep[23]   149.50    0.12 5.59  142.21  149.56  156.80  2063    1
## y_rep[24]   160.34    0.13 5.50  153.20  160.41  167.24  1891    1
## y_rep[25]   160.43    0.13 5.61  153.18  160.48  167.54  2009    1
## y_rep[26]   149.51    0.12 5.65  142.32  149.45  156.72  2061    1
## y_rep[27]   149.61    0.13 5.56  142.37  149.65  156.80  1916    1
## y_rep[28]   149.66    0.12 5.63  142.52  149.61  156.79  2049    1
## y_rep[29]   149.42    0.13 5.51  142.54  149.29  156.66  1904    1
## y_rep[30]   149.78    0.13 5.49  142.82  149.82  156.73  1805    1
## y_rep[31]   149.43    0.12 5.58  142.12  149.60  156.37  1998    1
## y_rep[32]   160.20    0.13 5.67  153.12  160.14  167.45  1879    1
## y_rep[33]   160.25    0.13 5.65  153.01  160.15  167.69  1979    1
## y_rep[34]   160.42    0.12 5.64  153.23  160.29  167.87  2040    1
## y_rep[35]   160.17    0.13 5.56  153.14  160.21  167.28  1731    1
## y_rep[36]   149.75    0.13 5.59  142.63  149.60  157.05  1991    1
## y_rep[37]   149.59    0.13 5.66  142.38  149.69  156.80  1889    1
## y_rep[38]   160.39    0.13 5.50  153.56  160.24  167.35  1890    1
## y_rep[39]   149.61    0.12 5.49  142.53  149.67  156.60  1964    1
## y_rep[40]   160.39    0.12 5.57  153.51  160.11  167.62  2130    1
## y_rep[41]   160.27    0.12 5.53  153.12  160.34  167.09  1983    1
## y_rep[42]   160.35    0.12 5.50  153.61  160.19  167.23  2101    1
## y_rep[43]   149.56    0.12 5.46  142.61  149.48  156.42  1978    1
## y_rep[44]   149.75    0.13 5.55  142.79  149.82  156.77  1902    1
## y_rep[45]   149.90    0.13 5.56  142.64  149.94  157.14  1710    1
## y_rep[46]   149.58    0.12 5.52  142.67  149.60  156.65  2015    1
## y_rep[47]   160.27    0.13 5.50  153.25  160.33  167.33  1733    1
## y_rep[48]   149.57    0.12 5.56  142.49  149.48  156.78  2062    1
## y_rep[49]   160.37    0.12 5.58  153.28  160.48  167.31  2047    1
## y_rep[50]   149.60    0.12 5.54  142.41  149.67  156.61  2026    1
## y_rep[51]   149.70    0.12 5.55  142.63  149.60  156.77  2069    1
## y_rep[52]   149.62    0.13 5.55  142.36  149.77  156.55  1943    1
## y_rep[53]   160.15    0.13 5.56  153.20  160.17  167.15  1932    1
## y_rep[54]   160.41    0.12 5.57  153.18  160.44  167.49  2000    1
## y_rep[55]   149.66    0.14 5.49  142.77  149.75  156.87  1467    1
## y_rep[56]   160.15    0.13 5.61  153.06  160.00  167.46  1802    1
## y_rep[57]   149.79    0.12 5.63  142.67  149.64  156.97  2203    1
## y_rep[58]   160.31    0.12 5.44  153.43  160.33  167.39  1977    1
## y_rep[59]   149.87    0.13 5.69  142.50  149.94  157.08  1920    1
## y_rep[60]   160.25    0.13 5.63  152.93  160.31  167.58  1983    1
## y_rep[61]   149.46    0.13 5.42  142.64  149.52  156.28  1759    1
## y_rep[62]   149.55    0.12 5.56  142.51  149.59  156.60  2016    1
## y_rep[63]   160.23    0.12 5.57  152.83  160.31  167.30  2094    1
## y_rep[64]   160.13    0.13 5.72  152.65  160.19  167.33  1884    1
## y_rep[65]   149.53    0.12 5.50  142.49  149.62  156.50  2186    1
## y_rep[66]   160.30    0.13 5.70  153.27  160.24  167.69  2028    1
## y_rep[67]   149.50    0.12 5.67  142.06  149.64  156.47  2114    1
## y_rep[68]   149.77    0.13 5.56  142.85  149.95  156.61  1942    1
## y_rep[69]   160.55    0.13 5.46  153.70  160.63  167.37  1860    1
## y_rep[70]   149.57    0.13 5.64  142.50  149.55  156.92  1931    1
## y_rep[71]   160.39    0.13 5.58  153.27  160.36  167.60  1908    1
## y_rep[72]   160.40    0.12 5.59  153.15  160.51  167.52  2005    1
## y_rep[73]   160.45    0.12 5.68  153.07  160.52  167.70  2181    1
## y_rep[74]   149.33    0.12 5.67  142.16  149.55  156.60  2094    1
## y_rep[75]   149.46    0.12 5.59  142.40  149.50  156.69  2044    1
## y_rep[76]   149.21    0.12 5.53  142.09  149.32  156.28  2036    1
## y_rep[77]   149.40    0.14 5.65  142.07  149.54  156.76  1686    1
## y_rep[78]   160.33    0.13 5.58  153.05  160.36  167.35  1778    1
## y_rep[79]   149.35    0.13 5.73  142.07  149.31  156.76  1822    1
## y_rep[80]   149.64    0.13 5.67  142.69  149.51  156.87  1948    1
## y_rep[81]   160.47    0.12 5.60  153.09  160.54  167.71  2067    1
## y_rep[82]   160.33    0.13 5.63  153.33  160.29  167.52  1759    1
## y_rep[83]   149.61    0.12 5.51  142.67  149.67  156.56  1965    1
## y_rep[84]   160.30    0.12 5.57  153.18  160.24  167.81  2112    1
## y_rep[85]   149.53    0.13 5.68  142.40  149.47  156.94  1910    1
## y_rep[86]   160.46    0.14 5.69  153.34  160.36  167.80  1745    1
## y_rep[87]   149.76    0.13 5.64  142.71  149.93  156.88  1888    1
## y_rep[88]   160.42    0.13 5.68  153.00  160.34  167.81  1950    1
## y_rep[89]   149.53    0.13 5.60  142.13  149.49  156.60  1925    1
## y_rep[90]   160.26    0.13 5.63  152.92  160.29  167.60  1965    1
## y_rep[91]   160.30    0.13 5.60  153.01  160.48  167.35  1849    1
## y_rep[92]   160.38    0.13 5.51  153.51  160.32  167.45  1942    1
## y_rep[93]   149.66    0.13 5.62  142.55  149.55  156.97  1986    1
## y_rep[94]   149.52    0.13 5.57  142.37  149.38  156.92  1861    1
## y_rep[95]   160.24    0.12 5.63  153.03  160.31  167.36  2055    1
## y_rep[96]   160.21    0.12 5.50  153.00  160.09  167.29  1960    1
## y_rep[97]   149.40    0.12 5.71  142.05  149.73  156.63  2113    1
## y_rep[98]   160.58    0.12 5.67  153.32  160.46  167.84  2058    1
## y_rep[99]   160.24    0.14 5.42  153.07  160.35  167.05  1586    1
## y_rep[100]  149.35    0.12 5.39  142.52  149.11  156.24  1878    1
## y_rep[101]  160.24    0.14 5.68  152.78  160.34  167.35  1670    1
## y_rep[102]  149.52    0.13 5.66  142.08  149.56  156.73  1886    1
## y_rep[103]  149.58    0.13 5.68  142.27  149.46  156.85  1815    1
## y_rep[104]  160.31    0.12 5.54  153.20  160.33  167.33  1993    1
## y_rep[105]  160.29    0.12 5.64  153.20  160.24  167.23  2097    1
## y_rep[106]  149.53    0.13 5.66  142.32  149.64  156.40  1916    1
## y_rep[107]  149.45    0.12 5.71  142.10  149.34  156.80  2093    1
## y_rep[108]  149.56    0.12 5.58  142.41  149.30  156.79  2020    1
## y_rep[109]  160.42    0.13 5.61  153.34  160.67  167.24  1973    1
## y_rep[110]  149.75    0.12 5.48  142.82  149.76  156.71  1947    1
## y_rep[111]  160.45    0.12 5.57  153.24  160.26  167.61  2044    1
## y_rep[112]  149.76    0.12 5.45  142.79  149.80  156.87  2045    1
## y_rep[113]  149.56    0.13 5.52  142.44  149.52  156.68  1851    1
## y_rep[114]  160.04    0.13 5.69  152.70  160.06  167.27  1992    1
## y_rep[115]  160.28    0.13 5.60  152.88  160.23  167.54  1783    1
## y_rep[116]  149.72    0.13 5.54  142.53  149.84  156.82  1891    1
## y_rep[117]  160.51    0.12 5.54  153.61  160.57  167.65  1976    1
## y_rep[118]  160.39    0.12 5.47  153.39  160.47  167.42  1979    1
## y_rep[119]  149.62    0.14 5.65  142.62  149.38  156.91  1751    1
## y_rep[120]  149.41    0.13 5.63  142.26  149.38  156.60  1954    1
## y_rep[121]  160.45    0.12 5.62  153.15  160.54  167.78  2079    1
## y_rep[122]  149.46    0.13 5.62  142.31  149.41  156.71  1914    1
## y_rep[123]  160.22    0.13 5.68  152.88  160.18  167.37  2022    1
## y_rep[124]  149.70    0.13 5.69  142.55  149.68  157.11  1904    1
## y_rep[125]  149.59    0.13 5.65  142.56  149.51  157.17  1957    1
## y_rep[126]  149.48    0.13 5.43  142.32  149.61  156.34  1731    1
## y_rep[127]  149.53    0.13 5.66  142.14  149.52  156.90  1961    1
## y_rep[128]  160.32    0.12 5.51  153.07  160.23  167.31  1950    1
## y_rep[129]  149.55    0.13 5.45  142.35  149.55  156.49  1731    1
## y_rep[130]  149.50    0.12 5.69  142.28  149.37  156.88  2246    1
## y_rep[131]  160.21    0.12 5.50  153.09  160.29  167.42  1939    1
## y_rep[132]  160.44    0.14 5.62  153.23  160.51  167.74  1702    1
## y_rep[133]  149.63    0.13 5.67  142.39  149.65  156.66  1810    1
## y_rep[134]  160.37    0.13 5.56  153.29  160.53  167.44  1916    1
## y_rep[135]  149.42    0.13 5.58  142.33  149.36  156.62  1857    1
## y_rep[136]  160.24    0.13 5.56  152.93  160.34  167.35  1960    1
## y_rep[137]  160.31    0.12 5.59  153.26  160.33  167.53  2100    1
## y_rep[138]  160.33    0.13 5.71  152.75  160.26  167.73  1978    1
## y_rep[139]  149.79    0.13 5.58  142.76  149.71  157.07  1931    1
## y_rep[140]  160.18    0.13 5.53  153.16  160.05  167.31  1904    1
## y_rep[141]  149.41    0.13 5.61  142.15  149.36  156.59  1931    1
## y_rep[142]  149.77    0.13 5.73  142.22  149.93  156.93  1817    1
## y_rep[143]  149.73    0.12 5.52  142.54  149.69  156.81  2126    1
## y_rep[144]  149.56    0.13 5.56  142.60  149.51  156.71  1946    1
## y_rep[145]  149.38    0.12 5.55  142.30  149.39  156.57  2086    1
## y_rep[146]  160.21    0.12 5.38  153.25  160.28  167.07  1980    1
## y_rep[147]  149.34    0.13 5.86  141.64  149.40  156.84  1905    1
## y_rep[148]  149.55    0.13 5.61  142.45  149.55  156.90  1919    1
## y_rep[149]  160.19    0.13 5.58  153.02  160.15  167.35  1965    1
## y_rep[150]  149.70    0.12 5.60  142.68  149.61  156.93  2127    1
## y_rep[151]  160.41    0.13 5.51  153.23  160.42  167.40  1939    1
## y_rep[152]  149.67    0.13 5.49  142.74  149.59  156.42  1901    1
## y_rep[153]  160.34    0.13 5.53  153.29  160.32  167.64  1860    1
## y_rep[154]  149.72    0.12 5.58  142.49  149.95  156.94  2165    1
## y_rep[155]  160.29    0.12 5.52  153.15  160.45  167.37  2199    1
## y_rep[156]  149.51    0.12 5.51  142.47  149.55  156.55  2039    1
## y_rep[157]  149.63    0.12 5.49  142.50  149.61  156.75  2135    1
## y_rep[158]  160.48    0.13 5.64  153.34  160.40  167.75  1902    1
## y_rep[159]  160.53    0.13 5.69  153.33  160.55  167.59  2023    1
## y_rep[160]  149.34    0.13 5.57  142.29  149.10  156.56  1927    1
## y_rep[161]  160.26    0.13 5.69  152.90  160.28  167.62  1987    1
## y_rep[162]  149.42    0.13 5.60  142.22  149.47  156.65  1897    1
## y_rep[163]  160.30    0.13 5.60  153.22  160.33  167.49  1891    1
## y_rep[164]  160.14    0.12 5.59  153.00  160.00  167.31  2068    1
## y_rep[165]  160.22    0.12 5.34  153.40  160.28  166.92  2060    1
## y_rep[166]  149.55    0.13 5.53  142.59  149.47  156.76  1742    1
## y_rep[167]  160.43    0.13 5.64  152.85  160.59  167.60  1847    1
## y_rep[168]  149.51    0.12 5.57  142.19  149.52  156.58  2040    1
## y_rep[169]  149.62    0.13 5.47  142.75  149.55  156.66  1880    1
## y_rep[170]  149.57    0.13 5.58  142.30  149.68  156.60  1954    1
## y_rep[171]  149.79    0.12 5.51  142.47  149.90  156.82  2115    1
## y_rep[172]  160.32    0.12 5.61  153.09  160.40  167.34  2026    1
## y_rep[173]  149.59    0.12 5.57  142.26  149.56  156.77  2041    1
## y_rep[174]  160.18    0.12 5.61  152.92  160.22  167.30  2039    1
## y_rep[175]  149.64    0.12 5.73  142.19  149.68  156.86  2114    1
## y_rep[176]  160.56    0.13 5.66  153.17  160.54  168.05  1923    1
## y_rep[177]  149.62    0.13 5.59  142.57  149.61  156.76  1904    1
## y_rep[178]  149.71    0.12 5.53  142.57  149.71  156.95  2017    1
## y_rep[179]  160.40    0.13 5.61  153.23  160.41  167.54  2001    1
## y_rep[180]  149.72    0.13 5.63  142.41  149.85  156.75  1902    1
## y_rep[181]  160.27    0.12 5.60  153.22  160.28  167.52  2013    1
## y_rep[182]  149.56    0.12 5.58  142.15  149.57  156.63  2228    1
## y_rep[183]  149.50    0.13 5.56  142.32  149.41  156.56  1733    1
## y_rep[184]  149.65    0.12 5.51  142.81  149.46  156.87  1980    1
## y_rep[185]  160.40    0.13 5.63  153.21  160.39  167.59  1961    1
## y_rep[186]  160.13    0.12 5.59  153.03  160.18  167.28  2082    1
## y_rep[187]  149.43    0.13 5.65  142.20  149.60  156.77  1981    1
## y_rep[188]  160.27    0.12 5.58  153.16  160.29  167.38  2036    1
## y_rep[189]  149.44    0.13 5.49  142.39  149.41  156.67  1829    1
## y_rep[190]  160.50    0.13 5.60  153.17  160.47  167.58  1958    1
## y_rep[191]  149.58    0.12 5.46  142.67  149.54  156.67  1965    1
## y_rep[192]  160.38    0.12 5.51  153.18  160.45  167.22  1992    1
## y_rep[193]  149.63    0.13 5.55  142.53  149.69  156.75  1907    1
## y_rep[194]  160.29    0.13 5.61  153.14  160.30  167.54  1880    1
## y_rep[195]  149.50    0.13 5.60  142.50  149.44  157.13  1903    1
## y_rep[196]  149.49    0.13 5.64  142.16  149.57  156.80  1945    1
## y_rep[197]  149.33    0.12 5.40  142.46  149.13  156.24  2019    1
## y_rep[198]  160.52    0.13 5.60  153.39  160.49  167.69  1962    1
## y_rep[199]  160.19    0.13 5.78  152.94  160.12  167.57  1847    1
## y_rep[200]  149.62    0.13 5.62  142.26  149.71  156.76  1997    1
## y_rep[201]  160.26    0.12 5.61  153.14  160.18  167.43  2088    1
## y_rep[202]  149.66    0.13 5.49  142.53  149.61  156.92  1844    1
## y_rep[203]  160.33    0.13 5.62  153.28  160.28  167.64  1768    1
## y_rep[204]  149.60    0.13 5.67  142.29  149.63  156.87  1948    1
## y_rep[205]  160.41    0.12 5.45  153.45  160.46  167.40  2034    1
## y_rep[206]  149.56    0.13 5.57  142.43  149.58  156.75  1921    1
## y_rep[207]  149.50    0.12 5.56  142.70  149.53  156.51  2213    1
## y_rep[208]  149.77    0.12 5.52  142.78  149.72  156.78  1959    1
## y_rep[209]  149.50    0.13 5.63  142.42  149.53  156.73  1879    1
## y_rep[210]  149.61    0.12 5.50  142.44  149.61  156.57  2007    1
## y_rep[211]  160.24    0.13 5.59  152.96  160.35  167.28  1977    1
## y_rep[212]  160.27    0.13 5.74  153.01  160.15  167.76  2018    1
## y_rep[213]  160.27    0.12 5.43  153.34  160.26  167.00  1926    1
## y_rep[214]  149.54    0.12 5.45  142.64  149.60  156.49  2018    1
## y_rep[215]  160.45    0.13 5.60  153.41  160.44  167.58  1913    1
## y_rep[216]  160.35    0.13 5.75  152.82  160.25  167.68  2012    1
## y_rep[217]  160.10    0.12 5.56  152.80  160.06  167.06  1981    1
## y_rep[218]  149.66    0.13 5.65  142.49  149.78  156.77  1961    1
## y_rep[219]  160.25    0.13 5.57  153.33  160.13  167.23  1828    1
## y_rep[220]  149.49    0.13 5.55  142.33  149.42  156.80  1703    1
## y_rep[221]  149.50    0.12 5.58  142.35  149.55  156.56  2247    1
## y_rep[222]  149.63    0.13 5.66  142.31  149.49  156.89  1986    1
## y_rep[223]  160.36    0.13 5.51  153.07  160.23  167.62  1915    1
## y_rep[224]  149.33    0.13 5.56  142.08  149.27  156.50  1901    1
## y_rep[225]  160.39    0.14 5.63  153.24  160.30  167.38  1662    1
## y_rep[226]  149.58    0.12 5.50  142.44  149.79  156.48  2083    1
## y_rep[227]  149.52    0.13 5.67  142.22  149.42  156.77  2002    1
## y_rep[228]  160.26    0.13 5.68  153.23  160.34  167.52  1897    1
## y_rep[229]  149.60    0.13 5.59  142.27  149.62  156.86  1882    1
## y_rep[230]  160.20    0.12 5.59  152.97  160.13  167.33  2059    1
## y_rep[231]  160.26    0.12 5.44  153.15  160.29  167.13  1954    1
## y_rep[232]  149.61    0.12 5.52  142.57  149.71  156.50  2117    1
## y_rep[233]  149.50    0.12 5.46  142.50  149.47  156.60  2069    1
## y_rep[234]  160.29    0.12 5.66  152.90  160.37  167.51  2152    1
## y_rep[235]  160.27    0.13 5.63  153.28  160.31  167.36  1812    1
## y_rep[236]  149.72    0.14 5.63  142.75  149.89  156.98  1690    1
## y_rep[237]  160.19    0.12 5.57  152.85  160.09  167.47  2257    1
## y_rep[238]  149.37    0.12 5.55  142.20  149.50  156.28  2088    1
## y_rep[239]  149.49    0.12 5.52  142.53  149.51  156.55  2037    1
## y_rep[240]  149.66    0.13 5.59  142.33  149.60  156.72  1878    1
## y_rep[241]  149.73    0.12 5.56  142.60  149.62  156.86  2108    1
## y_rep[242]  160.40    0.13 5.73  152.99  160.30  167.81  2012    1
## y_rep[243]  160.27    0.13 5.43  153.24  160.24  167.29  1795    1
## y_rep[244]  149.61    0.13 5.60  142.65  149.38  156.91  1976    1
## y_rep[245]  160.28    0.12 5.54  153.25  160.10  167.37  2114    1
## y_rep[246]  149.55    0.14 5.68  142.05  149.47  156.81  1753    1
## y_rep[247]  160.18    0.13 5.55  153.19  160.22  167.42  1797    1
## y_rep[248]  149.48    0.13 5.62  142.46  149.58  156.70  1923    1
## y_rep[249]  160.41    0.13 5.69  153.05  160.33  167.59  2071    1
## y_rep[250]  160.22    0.13 5.57  153.28  160.26  167.26  1926    1
## y_rep[251]  160.50    0.13 5.53  153.46  160.41  167.63  1753    1
## y_rep[252]  160.01    0.13 5.67  152.96  159.95  167.28  1971    1
## y_rep[253]  149.52    0.13 5.55  142.38  149.55  156.70  1843    1
## y_rep[254]  149.66    0.13 5.57  142.43  149.58  156.92  1931    1
## y_rep[255]  149.45    0.13 5.63  142.16  149.46  156.53  1850    1
## y_rep[256]  149.66    0.13 5.48  142.78  149.68  156.52  1877    1
## y_rep[257]  160.20    0.12 5.49  153.38  160.13  167.30  2031    1
## y_rep[258]  149.47    0.13 5.63  142.23  149.54  156.45  2025    1
## y_rep[259]  160.24    0.12 5.48  153.26  160.31  167.20  2122    1
## y_rep[260]  160.33    0.13 5.70  153.20  160.07  167.57  2027    1
## y_rep[261]  160.31    0.13 5.56  153.13  160.34  167.27  1930    1
## y_rep[262]  149.58    0.12 5.43  142.50  149.53  156.35  1920    1
## y_rep[263]  149.54    0.13 5.69  142.25  149.41  156.63  1963    1
## y_rep[264]  149.35    0.12 5.54  141.99  149.50  156.46  2177    1
## y_rep[265]  160.16    0.13 5.54  152.89  160.29  167.24  1866    1
## y_rep[266]  149.70    0.13 5.54  142.67  149.65  156.85  1810    1
## y_rep[267]  160.05    0.12 5.43  153.19  160.04  167.22  1914    1
## y_rep[268]  149.56    0.13 5.53  142.51  149.60  156.53  1762    1
## y_rep[269]  149.45    0.13 5.60  142.33  149.45  156.47  1917    1
## y_rep[270]  160.05    0.13 5.56  152.94  160.06  167.10  1722    1
## y_rep[271]  160.42    0.13 5.61  153.20  160.38  167.62  1800    1
## y_rep[272]  149.41    0.13 5.75  142.05  149.53  156.63  2059    1
## y_rep[273]  160.25    0.13 5.61  152.99  160.28  167.27  2013    1
## y_rep[274]  160.14    0.14 5.37  153.32  160.30  166.85  1563    1
## y_rep[275]  149.46    0.14 5.38  142.74  149.42  156.34  1545    1
## y_rep[276]  160.16    0.13 5.62  152.87  160.23  167.26  1839    1
## y_rep[277]  149.68    0.12 5.61  142.37  149.70  156.92  2094    1
## y_rep[278]  149.63    0.12 5.54  142.50  149.85  156.61  2169    1
## y_rep[279]  160.32    0.12 5.67  153.02  160.36  167.48  2134    1
## y_rep[280]  149.67    0.13 5.58  142.43  149.67  156.70  1906    1
## y_rep[281]  149.47    0.13 5.48  142.46  149.25  156.72  1904    1
## y_rep[282]  160.62    0.13 5.69  153.38  160.46  168.03  2067    1
## y_rep[283]  149.39    0.13 5.64  142.09  149.38  156.62  2005    1
## y_rep[284]  160.37    0.13 5.63  153.32  160.23  167.62  1833    1
## y_rep[285]  149.50    0.12 5.45  142.60  149.46  156.49  2071    1
## y_rep[286]  160.31    0.12 5.65  152.96  160.39  167.60  2062    1
## y_rep[287]  160.33    0.13 5.53  153.25  160.33  167.42  1851    1
## y_rep[288]  160.30    0.12 5.50  153.25  160.35  167.10  2166    1
## y_rep[289]  149.47    0.13 5.62  142.10  149.55  156.54  1910    1
## y_rep[290]  160.23    0.13 5.48  153.21  160.14  167.46  1747    1
## y_rep[291]  160.28    0.13 5.61  153.12  160.32  167.44  1959    1
## y_rep[292]  160.30    0.13 5.54  153.44  160.16  167.68  1961    1
## y_rep[293]  160.45    0.13 5.65  153.41  160.13  167.89  2025    1
## y_rep[294]  149.35    0.13 5.54  142.14  149.35  156.20  1958    1
## y_rep[295]  160.27    0.12 5.42  153.40  160.20  167.12  1979    1
## y_rep[296]  149.51    0.12 5.58  142.23  149.56  156.82  2030    1
## y_rep[297]  149.68    0.13 5.54  142.74  149.48  156.90  1958    1
## y_rep[298]  149.51    0.13 5.51  142.42  149.49  156.48  1931    1
## y_rep[299]  160.11    0.13 5.44  153.19  159.93  167.37  1786    1
## y_rep[300]  160.34    0.12 5.55  153.13  160.27  167.36  2065    1
## y_rep[301]  149.55    0.13 5.59  142.42  149.46  156.73  1822    1
## y_rep[302]  160.24    0.12 5.54  152.94  160.30  167.29  2160    1
## y_rep[303]  149.69    0.13 5.51  142.50  149.60  156.75  1713    1
## y_rep[304]  160.41    0.12 5.40  153.51  160.61  167.07  1965    1
## y_rep[305]  160.35    0.13 5.62  153.21  160.28  167.64  1912    1
## y_rep[306]  160.13    0.12 5.43  153.17  160.19  167.29  1946    1
## y_rep[307]  160.29    0.12 5.44  153.13  160.38  167.25  1965    1
## y_rep[308]  149.70    0.12 5.50  142.66  149.68  156.79  1993    1
## y_rep[309]  149.70    0.12 5.57  142.37  149.67  156.95  2053    1
## y_rep[310]  149.50    0.13 5.56  142.63  149.41  156.74  1836    1
## y_rep[311]  149.58    0.13 5.66  142.23  149.59  156.80  1803    1
## y_rep[312]  149.42    0.13 5.49  142.42  149.46  156.56  1854    1
## y_rep[313]  160.51    0.13 5.80  153.06  160.52  167.91  2051    1
## y_rep[314]  149.38    0.13 5.64  142.24  149.27  156.52  1929    1
## y_rep[315]  160.20    0.13 5.47  153.15  160.29  167.25  1736    1
## y_rep[316]  149.52    0.12 5.43  142.62  149.48  156.51  1994    1
## y_rep[317]  160.16    0.13 5.45  153.12  160.21  166.99  1858    1
## y_rep[318]  149.47    0.13 5.70  142.36  149.44  156.95  1925    1
## y_rep[319]  160.47    0.13 5.64  152.99  160.54  167.54  2015    1
## y_rep[320]  149.52    0.13 5.51  142.73  149.61  156.61  1919    1
## y_rep[321]  160.36    0.13 5.69  153.16  160.27  167.85  1971    1
## y_rep[322]  149.60    0.12 5.39  142.85  149.63  156.46  1977    1
## y_rep[323]  160.47    0.12 5.50  153.26  160.47  167.61  2197    1
## y_rep[324]  160.52    0.13 5.56  153.26  160.52  167.61  1939    1
## y_rep[325]  149.41    0.14 5.53  142.05  149.23  156.34  1655    1
## y_rep[326]  149.54    0.13 5.69  142.55  149.51  156.90  1889    1
## y_rep[327]  149.74    0.12 5.53  142.70  149.88  156.61  1969    1
## y_rep[328]  160.40    0.13 5.50  153.24  160.43  167.25  1859    1
## y_rep[329]  160.30    0.13 5.70  153.07  160.39  167.59  2049    1
## y_rep[330]  160.29    0.13 5.55  153.27  160.27  167.47  1847    1
## y_rep[331]  160.32    0.13 5.71  153.06  160.42  167.47  1971    1
## y_rep[332]  160.24    0.12 5.53  153.06  160.27  167.38  2120    1
## y_rep[333]  149.46    0.13 5.56  142.29  149.44  156.43  1903    1
## y_rep[334]  160.24    0.13 5.55  153.15  160.08  167.40  1884    1
## y_rep[335]  149.53    0.12 5.53  142.40  149.59  156.58  2095    1
## y_rep[336]  160.42    0.13 5.75  152.92  160.42  167.61  1865    1
## y_rep[337]  149.52    0.12 5.47  142.65  149.47  156.63  2130    1
## y_rep[338]  149.59    0.13 5.61  142.31  149.67  156.89  2017    1
## y_rep[339]  149.72    0.12 5.50  142.73  149.65  156.72  1985    1
## y_rep[340]  149.37    0.12 5.49  142.30  149.50  156.22  1932    1
## y_rep[341]  149.70    0.12 5.50  142.79  149.75  156.80  1946    1
## y_rep[342]  160.14    0.12 5.49  153.08  160.18  167.26  2038    1
## y_rep[343]  149.36    0.13 5.59  142.30  149.30  156.45  1909    1
## y_rep[344]  160.44    0.13 5.48  153.40  160.57  167.25  1847    1
## y_rep[345]  149.65    0.13 5.61  142.64  149.68  156.73  1826    1
## y_rep[346]  160.38    0.12 5.58  153.37  160.32  167.49  1996    1
## y_rep[347]  149.71    0.14 5.68  142.59  149.56  157.32  1692    1
## y_rep[348]  149.42    0.12 5.55  142.39  149.45  156.49  1994    1
## y_rep[349]  149.51    0.12 5.51  142.52  149.53  156.48  2011    1
## y_rep[350]  160.22    0.13 5.75  152.74  160.25  167.46  2093    1
## y_rep[351]  149.74    0.14 5.60  142.46  149.84  156.91  1513    1
## y_rep[352]  160.17    0.13 5.69  152.88  160.17  167.61  1844    1
## lp__       -782.21    0.04 1.17 -783.79 -781.92 -781.02   866    1
## 
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:56:38 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

5 Polynomial regression

5.1 Howell 2: subset with only individuals over 18 years old

Standardizing the explanatory variable:

howell2 <- howell2 %>% mutate(weight.z = (weight - mean(weight)) / sd(weight))

Data in a list:

m6.data <- list(N = nrow(howell2), height = howell2$height, weight = howell2$weight.z)
data {                                                     // observed variables 
  int<lower=1> N;                                          // number of observations
  vector[N] height;                                        // response variables
  vector[N] weight;                                        // predictor (previously centered)
}

parameters {                                               // unobserved variables
  real beta[3];                                            // parameters: intercept and slope
  real sigma;
}

model {
  height ~ normal(beta[1] + beta[2]*weight + beta[3]*(weight .* weight),sigma);        // likelihood
  beta[1] ~ normal(178, 100);                              // prior of the intercept
  beta[2] ~ normal(0, 100);                                  // prior of the slope
  beta[3] ~ normal(0, 100);
  sigma ~ uniform(0,50);
}

generated quantities {
  vector[N] y_rep;

  for(i in 1:N)
  y_rep[i] = normal_rng(beta[1] + beta[2]*weight[i] + beta[3]*(weight[i] .* weight[i]), sigma);
}
fit.m6 <- sampling(m6, data = m6.data, iter = 1000, chains = 4, cores = 4)
print(fit.m6, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: c7fbeba14f23c52987bbbb1ee94a3a29.
## 4 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=2000.
## 
##               mean se_mean   sd     10%     50%     90% n_eff Rhat
## beta[1]     154.61    0.01 0.36  154.15  154.61  155.07  1018 1.00
## beta[2]       5.85    0.01 0.27    5.51    5.86    6.20   799 1.00
## beta[3]      -0.01    0.01 0.22   -0.29   -0.02    0.29   988 1.00
## sigma         5.11    0.00 0.19    4.86    5.11    5.35  2249 1.00
## y_rep[1]    157.25    0.11 5.19  150.72  157.24  163.86  2041 1.00
## y_rep[2]    146.94    0.11 5.07  140.51  146.86  153.45  1995 1.00
## y_rep[3]    142.78    0.12 5.20  135.99  142.81  149.52  1889 1.00
## y_rep[4]    161.85    0.12 5.14  155.16  161.72  168.38  1922 1.00
## y_rep[5]    151.37    0.11 5.10  144.74  151.47  157.58  2054 1.00
## y_rep[6]    170.67    0.13 5.44  163.83  170.58  177.49  1839 1.00
## y_rep[7]    148.26    0.12 5.15  141.63  148.17  154.92  1964 1.00
## y_rep[8]    164.10    0.12 5.36  157.28  164.15  170.81  2072 1.00
## y_rep[9]    145.60    0.14 5.27  138.80  145.65  152.30  1488 1.00
## y_rep[10]   163.15    0.12 5.16  156.48  163.23  169.62  1864 1.00
## y_rep[11]   159.00    0.11 5.14  152.53  158.95  165.47  2010 1.00
## y_rep[12]   151.00    0.12 4.93  144.76  150.94  157.60  1793 1.00
## y_rep[13]   146.66    0.12 5.11  140.38  146.39  153.40  1794 1.00
## y_rep[14]   157.26    0.12 5.23  150.44  157.38  164.06  1880 1.00
## y_rep[15]   144.38    0.12 5.19  137.47  144.58  150.91  1735 1.00
## y_rep[16]   157.90    0.11 5.13  151.33  157.97  164.51  2031 1.00
## y_rep[17]   152.10    0.12 5.23  145.43  152.05  158.93  2056 1.00
## y_rep[18]   148.43    0.12 5.19  141.73  148.61  155.05  1819 1.00
## y_rep[19]   158.27    0.12 5.18  151.63  158.26  164.72  1868 1.00
## y_rep[20]   152.75    0.13 5.24  146.04  152.81  159.40  1705 1.00
## y_rep[21]   145.87    0.11 4.98  139.40  145.92  152.12  1972 1.00
## y_rep[22]   148.24    0.11 5.14  141.53  148.24  154.90  2086 1.00
## y_rep[23]   146.09    0.12 5.11  139.69  146.11  152.79  1871 1.00
## y_rep[24]   150.29    0.12 5.26  143.52  150.16  157.00  1927 1.00
## y_rep[25]   163.76    0.11 5.09  157.31  163.75  170.35  2004 1.00
## y_rep[26]   147.85    0.11 5.13  141.35  147.74  154.31  1994 1.00
## y_rep[27]   148.78    0.12 5.21  142.38  148.76  155.28  2043 1.00
## y_rep[28]   156.10    0.11 4.95  149.77  156.12  162.42  2005 1.00
## y_rep[29]   148.80    0.12 5.20  141.98  148.84  155.20  1996 1.00
## y_rep[30]   146.01    0.12 5.17  139.14  146.14  152.38  1861 1.00
## y_rep[31]   156.62    0.13 5.27  149.85  156.66  163.40  1540 1.01
## y_rep[32]   157.04    0.11 5.09  150.65  156.91  163.45  2011 1.00
## y_rep[33]   158.49    0.12 5.19  151.85  158.50  165.07  1925 1.00
## y_rep[34]   158.64    0.11 5.07  152.08  158.72  165.00  2104 1.00
## y_rep[35]   165.13    0.12 5.08  158.80  165.22  171.70  1930 1.00
## y_rep[36]   149.19    0.12 5.12  142.60  149.22  155.70  1966 1.00
## y_rep[37]   158.96    0.13 5.18  152.22  158.98  165.50  1680 1.00
## y_rep[38]   151.24    0.13 5.05  144.58  151.39  157.91  1541 1.00
## y_rep[39]   148.63    0.11 5.22  141.89  148.63  155.24  2254 1.00
## y_rep[40]   152.29    0.12 5.15  145.86  152.30  158.85  1906 1.00
## y_rep[41]   154.21    0.11 5.01  147.80  154.12  160.88  2018 1.00
## y_rep[42]   166.70    0.12 5.02  160.24  166.68  173.07  1654 1.00
## y_rep[43]   156.15    0.11 5.23  149.38  156.10  162.68  2101 1.00
## y_rep[44]   153.94    0.12 5.32  146.89  153.98  160.75  1914 1.00
## y_rep[45]   160.15    0.11 4.98  153.69  160.29  166.48  2004 1.00
## y_rep[46]   163.17    0.11 5.24  156.57  163.20  169.76  2184 1.00
## y_rep[47]   155.42    0.12 5.25  148.67  155.38  162.18  2047 1.00
## y_rep[48]   157.20    0.11 5.22  150.67  157.08  164.15  2108 1.00
## y_rep[49]   161.19    0.12 5.24  154.26  161.07  167.99  1964 1.00
## y_rep[50]   156.85    0.11 5.04  150.44  156.89  163.32  2051 1.00
## y_rep[51]   155.31    0.11 5.06  148.79  155.36  162.02  2007 1.00
## y_rep[52]   152.05    0.11 5.08  145.62  151.92  158.66  2014 1.00
## y_rep[53]   146.84    0.12 5.10  140.35  146.81  153.32  1963 1.00
## y_rep[54]   164.49    0.13 5.16  157.85  164.32  171.18  1659 1.00
## y_rep[55]   147.58    0.11 5.09  141.25  147.61  154.03  2131 1.00
## y_rep[56]   157.77    0.12 5.16  151.28  157.69  164.10  1943 1.00
## y_rep[57]   147.58    0.11 5.09  140.98  147.56  154.14  2035 1.00
## y_rep[58]   156.71    0.11 5.20  149.88  156.70  163.53  2088 1.00
## y_rep[59]   156.67    0.12 5.26  150.04  156.53  163.54  1941 1.00
## y_rep[60]   165.84    0.12 5.13  159.06  165.93  172.49  1930 1.00
## y_rep[61]   148.22    0.11 5.01  141.70  148.36  154.63  1939 1.00
## y_rep[62]   156.51    0.12 5.27  149.87  156.62  163.09  1945 1.00
## y_rep[63]   152.89    0.11 5.17  146.33  152.98  159.30  2034 1.00
## y_rep[64]   159.46    0.11 5.21  152.75  159.44  166.41  2199 1.00
## y_rep[65]   149.74    0.12 5.11  143.32  149.65  156.47  1837 1.00
## y_rep[66]   159.95    0.11 5.08  153.41  159.93  166.63  2024 1.00
## y_rep[67]   149.84    0.12 5.24  143.16  149.79  156.32  1979 1.00
## y_rep[68]   146.88    0.12 5.07  140.42  146.81  153.37  1857 1.00
## y_rep[69]   155.69    0.12 5.11  149.22  155.75  162.28  1963 1.00
## y_rep[70]   151.98    0.12 5.08  145.54  151.88  158.59  1858 1.00
## y_rep[71]   157.11    0.12 5.21  150.60  157.15  163.89  2046 1.00
## y_rep[72]   164.35    0.12 5.23  157.52  164.35  170.98  1839 1.00
## y_rep[73]   157.84    0.12 5.12  151.33  157.92  164.31  1949 1.00
## y_rep[74]   152.66    0.12 5.18  145.83  152.63  159.53  1947 1.00
## y_rep[75]   145.41    0.11 5.09  138.88  145.48  151.85  2159 1.00
## y_rep[76]   165.21    0.11 5.12  158.50  165.33  171.75  2015 1.00
## y_rep[77]   160.52    0.11 4.92  154.28  160.44  166.69  1870 1.00
## y_rep[78]   156.41    0.12 5.20  149.64  156.49  162.88  1938 1.00
## y_rep[79]   150.78    0.11 4.96  144.55  150.83  157.18  1939 1.00
## y_rep[80]   143.78    0.12 5.29  136.98  143.76  150.43  1891 1.00
## y_rep[81]   165.48    0.11 4.99  159.16  165.37  172.09  1969 1.00
## y_rep[82]   152.73    0.12 5.09  146.11  152.88  159.12  1916 1.00
## y_rep[83]   150.09    0.12 5.01  143.68  150.03  156.48  1776 1.00
## y_rep[84]   155.60    0.12 5.18  148.97  155.61  162.01  1994 1.00
## y_rep[85]   150.99    0.12 5.06  144.52  151.00  157.20  1842 1.00
## y_rep[86]   157.18    0.11 5.02  150.73  157.19  163.37  2130 1.00
## y_rep[87]   147.85    0.12 5.13  141.23  147.95  154.38  1788 1.00
## y_rep[88]   154.64    0.12 5.13  148.05  154.63  161.32  1944 1.00
## y_rep[89]   152.22    0.12 5.25  145.52  152.19  158.99  1935 1.00
## y_rep[90]   163.48    0.11 5.00  157.10  163.51  169.79  1987 1.00
## y_rep[91]   162.50    0.11 5.01  156.17  162.67  168.87  1973 1.00
## y_rep[92]   161.13    0.11 5.15  154.58  161.23  167.59  2222 1.00
## y_rep[93]   152.26    0.11 5.05  145.52  152.38  158.57  2100 1.00
## y_rep[94]   158.71    0.11 5.08  152.23  158.86  165.05  2179 1.00
## y_rep[95]   151.73    0.12 5.03  145.11  151.77  158.08  1842 1.00
## y_rep[96]   164.60    0.12 5.12  158.01  164.49  171.32  1949 1.00
## y_rep[97]   152.03    0.12 5.19  145.30  151.97  158.64  2015 1.00
## y_rep[98]   162.11    0.12 5.23  155.38  162.30  168.71  1986 1.00
## y_rep[99]   159.41    0.12 5.15  152.88  159.26  166.06  1979 1.00
## y_rep[100]  152.32    0.11 5.13  145.75  152.40  158.94  2050 1.00
## y_rep[101]  158.48    0.12 4.97  152.17  158.52  164.95  1861 1.00
## y_rep[102]  148.80    0.11 5.19  142.15  148.88  155.62  2248 1.00
## y_rep[103]  158.87    0.12 5.23  151.99  158.89  165.46  1989 1.00
## y_rep[104]  168.06    0.13 5.41  161.22  168.02  174.95  1808 1.00
## y_rep[105]  157.24    0.11 5.04  150.90  157.20  163.67  1920 1.00
## y_rep[106]  149.35    0.11 5.12  142.85  149.33  155.99  2045 1.00
## y_rep[107]  147.36    0.11 5.20  140.61  147.36  154.12  2082 1.00
## y_rep[108]  152.18    0.12 5.15  145.60  152.05  159.04  1928 1.00
## y_rep[109]  154.22    0.11 4.98  147.89  154.06  160.87  1918 1.00
## y_rep[110]  157.13    0.12 5.12  150.48  157.03  163.61  1764 1.00
## y_rep[111]  163.79    0.11 5.07  157.22  163.87  170.34  2014 1.00
## y_rep[112]  143.76    0.11 5.12  137.24  143.61  150.35  2001 1.00
## y_rep[113]  150.76    0.11 5.05  144.44  150.66  157.06  1966 1.00
## y_rep[114]  160.09    0.12 5.00  153.61  160.04  166.49  1783 1.00
## y_rep[115]  158.31    0.12 5.09  151.96  158.39  164.79  1861 1.00
## y_rep[116]  162.36    0.12 5.14  156.14  162.16  168.72  1937 1.00
## y_rep[117]  162.90    0.12 5.12  156.32  162.99  169.37  1899 1.00
## y_rep[118]  164.27    0.11 5.15  157.64  164.37  170.72  2067 1.00
## y_rep[119]  152.15    0.12 5.15  145.58  152.06  158.81  1910 1.00
## y_rep[120]  151.05    0.12 5.08  144.39  151.13  157.66  1925 1.00
## y_rep[121]  158.99    0.11 5.12  152.33  158.97  165.45  2138 1.00
## y_rep[122]  154.19    0.12 5.04  147.88  154.21  160.77  1728 1.00
## y_rep[123]  155.60    0.11 5.00  149.17  155.53  162.07  1981 1.00
## y_rep[124]  151.98    0.11 4.98  145.77  151.91  158.45  2127 1.00
## y_rep[125]  160.24    0.11 5.08  153.87  160.38  166.64  2129 1.00
## y_rep[126]  153.69    0.11 5.18  146.95  153.67  160.30  2101 1.00
## y_rep[127]  142.90    0.12 5.11  136.22  143.03  149.37  1894 1.00
## y_rep[128]  165.34    0.12 5.15  158.92  165.29  171.88  1981 1.00
## y_rep[129]  161.39    0.12 5.31  154.70  161.36  168.07  1847 1.00
## y_rep[130]  147.10    0.11 5.15  140.53  146.98  153.64  2061 1.00
## y_rep[131]  158.13    0.11 5.15  151.57  158.11  164.66  2065 1.00
## y_rep[132]  165.50    0.12 5.15  158.87  165.59  171.97  1865 1.00
## y_rep[133]  151.97    0.11 5.04  145.62  151.99  158.39  1927 1.00
## y_rep[134]  159.36    0.11 4.94  152.92  159.37  165.69  1851 1.00
## y_rep[135]  156.02    0.11 5.08  149.69  155.89  162.62  2087 1.00
## y_rep[136]  169.86    0.12 5.36  162.99  170.00  176.53  1859 1.00
## y_rep[137]  158.20    0.12 5.19  151.67  158.18  164.82  1947 1.00
## y_rep[138]  142.54    0.11 5.17  136.18  142.49  149.07  2022 1.00
## y_rep[139]  145.65    0.12 5.16  138.97  145.57  152.32  1774 1.00
## y_rep[140]  155.17    0.11 4.91  148.99  155.14  161.44  1868 1.00
## y_rep[141]  153.67    0.11 5.11  147.15  153.77  160.09  2034 1.00
## y_rep[142]  155.17    0.11 5.05  148.58  155.12  161.58  2020 1.00
## y_rep[143]  149.24    0.12 4.99  142.79  149.14  155.70  1864 1.00
## y_rep[144]  151.06    0.11 5.03  144.87  151.07  157.37  1975 1.00
## y_rep[145]  154.80    0.11 5.04  148.32  154.76  161.32  1931 1.00
## y_rep[146]  162.38    0.11 5.00  155.80  162.26  169.00  1972 1.00
## y_rep[147]  161.26    0.12 5.22  154.66  161.40  167.83  1981 1.00
## y_rep[148]  149.32    0.12 5.11  142.86  149.28  155.93  1718 1.00
## y_rep[149]  155.22    0.12 5.21  148.40  155.38  161.91  1969 1.00
## y_rep[150]  150.55    0.12 5.10  143.72  150.74  156.86  1957 1.00
## y_rep[151]  160.25    0.12 5.16  153.69  160.26  166.81  1812 1.00
## y_rep[152]  148.79    0.12 5.02  142.56  148.88  155.15  1904 1.00
## y_rep[153]  149.72    0.12 5.26  142.97  149.77  156.34  2006 1.00
## y_rep[154]  154.02    0.12 5.17  147.27  154.02  160.62  2001 1.00
## y_rep[155]  149.76    0.11 5.14  143.22  149.87  156.16  2053 1.00
## y_rep[156]  141.62    0.12 5.45  134.58  141.66  148.54  1949 1.00
## y_rep[157]  156.18    0.11 5.11  149.63  156.11  162.78  2059 1.00
## y_rep[158]  150.49    0.13 5.25  143.47  150.56  157.10  1713 1.00
## y_rep[159]  159.36    0.11 5.12  152.65  159.24  165.74  2183 1.00
## y_rep[160]  151.34    0.11 4.94  145.11  151.39  157.77  1948 1.00
## y_rep[161]  163.45    0.12 5.23  156.57  163.44  170.23  1960 1.00
## y_rep[162]  154.69    0.11 5.13  148.17  154.62  161.18  2087 1.00
## y_rep[163]  146.31    0.12 5.11  139.70  146.32  152.96  1891 1.00
## y_rep[164]  154.81    0.11 5.15  148.14  154.90  161.69  2124 1.00
## y_rep[165]  157.42    0.12 5.17  150.97  157.27  163.96  1894 1.00
## y_rep[166]  155.35    0.12 5.27  148.36  155.38  162.06  1890 1.00
## y_rep[167]  157.70    0.11 5.05  151.24  157.67  164.19  2107 1.00
## y_rep[168]  151.11    0.12 5.22  144.54  150.89  157.79  2039 1.00
## y_rep[169]  148.68    0.11 5.13  142.27  148.58  155.19  2122 1.00
## y_rep[170]  152.02    0.12 4.98  145.51  152.10  158.43  1791 1.00
## y_rep[171]  148.50    0.12 5.14  141.92  148.40  155.01  1883 1.00
## y_rep[172]  157.69    0.12 5.23  151.19  157.69  164.33  1776 1.00
## y_rep[173]  149.10    0.12 5.15  142.53  149.09  155.74  1946 1.00
## y_rep[174]  150.02    0.11 5.14  143.55  149.89  156.60  2046 1.00
## y_rep[175]  159.37    0.11 5.12  152.65  159.44  165.84  2074 1.00
## y_rep[176]  162.96    0.12 5.22  156.35  163.02  169.60  1850 1.00
## y_rep[177]  155.29    0.12 5.20  148.53  155.33  162.13  1801 1.00
## y_rep[178]  149.80    0.12 5.23  143.14  149.97  156.38  1913 1.00
## y_rep[179]  152.87    0.12 5.24  145.94  152.86  159.72  1910 1.00
## y_rep[180]  142.69    0.12 5.21  135.76  142.70  149.20  2023 1.00
## y_rep[181]  155.02    0.12 5.15  148.47  154.93  161.56  1913 1.00
## y_rep[182]  148.22    0.12 5.00  141.81  148.23  154.58  1699 1.00
## y_rep[183]  146.39    0.12 5.13  139.78  146.39  152.95  1924 1.00
## y_rep[184]  150.84    0.12 5.09  144.26  150.80  157.42  1851 1.00
## y_rep[185]  157.21    0.11 5.10  150.53  157.31  163.27  1981 1.00
## y_rep[186]  157.01    0.12 5.04  150.63  156.92  163.58  1799 1.00
## y_rep[187]  155.76    0.11 5.07  149.32  155.57  162.58  2035 1.00
## y_rep[188]  151.14    0.11 5.16  144.50  151.12  157.79  2035 1.00
## y_rep[189]  146.94    0.12 5.18  140.40  146.92  153.73  1955 1.00
## y_rep[190]  157.24    0.11 5.09  150.54  157.30  163.50  2033 1.00
## y_rep[191]  146.95    0.12 5.11  140.43  147.02  153.46  1880 1.00
## y_rep[192]  165.00    0.12 5.17  158.38  165.03  171.73  1999 1.00
## y_rep[193]  148.75    0.12 5.22  142.12  148.66  155.61  1945 1.00
## y_rep[194]  156.90    0.12 5.18  150.24  156.85  163.48  1879 1.00
## y_rep[195]  146.65    0.11 5.05  140.12  146.64  153.24  1954 1.00
## y_rep[196]  151.32    0.11 5.13  144.81  151.43  158.01  2124 1.00
## y_rep[197]  154.38    0.12 5.19  147.70  154.51  161.19  1910 1.00
## y_rep[198]  159.46    0.12 5.21  152.90  159.44  166.14  1917 1.00
## y_rep[199]  164.02    0.12 5.21  157.12  163.98  170.85  1952 1.00
## y_rep[200]  148.11    0.11 5.00  141.63  148.18  154.37  2064 1.00
## y_rep[201]  166.73    0.12 5.31  159.93  166.74  173.62  1933 1.00
## y_rep[202]  154.04    0.12 5.21  147.34  154.17  160.70  1938 1.00
## y_rep[203]  153.97    0.12 5.29  147.06  154.08  160.71  1874 1.00
## y_rep[204]  154.21    0.11 5.06  147.96  154.22  160.72  1973 1.00
## y_rep[205]  154.05    0.12 5.11  147.54  153.96  160.67  1855 1.00
## y_rep[206]  150.28    0.11 5.03  143.80  150.33  156.71  1960 1.00
## y_rep[207]  153.87    0.12 5.03  147.27  153.87  160.41  1827 1.00
## y_rep[208]  145.06    0.11 5.13  138.42  145.08  151.60  2004 1.00
## y_rep[209]  143.50    0.12 5.16  136.83  143.44  150.20  2007 1.00
## y_rep[210]  146.43    0.12 5.12  139.89  146.40  152.89  1851 1.00
## y_rep[211]  158.72    0.12 5.12  152.08  158.93  165.26  1963 1.00
## y_rep[212]  162.17    0.12 5.07  155.91  162.10  168.68  1903 1.00
## y_rep[213]  147.61    0.12 5.05  141.17  147.61  154.02  1892 1.00
## y_rep[214]  150.69    0.11 5.08  143.96  150.79  157.01  2004 1.00
## y_rep[215]  159.78    0.12 5.09  153.42  159.74  166.30  1752 1.00
## y_rep[216]  153.71    0.12 5.26  147.03  153.76  160.37  1980 1.00
## y_rep[217]  159.19    0.12 5.16  152.38  159.36  165.75  2009 1.00
## y_rep[218]  144.38    0.11 5.21  137.67  144.34  150.97  2245 1.00
## y_rep[219]  153.59    0.11 5.06  147.12  153.46  160.00  1984 1.00
## y_rep[220]  157.53    0.12 5.24  151.41  157.38  164.24  1889 1.00
## y_rep[221]  152.42    0.11 5.06  145.88  152.39  159.20  2063 1.00
## y_rep[222]  144.50    0.12 5.21  137.76  144.54  151.16  1932 1.00
## y_rep[223]  150.00    0.12 5.12  143.41  149.90  156.45  1874 1.00
## y_rep[224]  146.53    0.11 5.15  139.95  146.66  153.10  2146 1.00
## y_rep[225]  160.08    0.12 5.30  153.27  160.20  166.81  1842 1.00
## y_rep[226]  155.14    0.12 5.08  148.56  155.30  161.48  1920 1.00
## y_rep[227]  158.45    0.12 4.98  152.02  158.63  164.86  1870 1.00
## y_rep[228]  167.41    0.12 5.16  160.98  167.46  174.02  1975 1.00
## y_rep[229]  153.09    0.12 5.08  146.62  153.10  159.58  1660 1.00
## y_rep[230]  161.01    0.12 5.09  154.56  161.01  167.59  1868 1.00
## y_rep[231]  159.69    0.12 5.18  153.13  159.58  166.53  1999 1.00
## y_rep[232]  145.00    0.12 5.19  138.30  145.01  151.71  1990 1.00
## y_rep[233]  149.50    0.12 5.18  142.79  149.53  156.23  1979 1.00
## y_rep[234]  167.78    0.12 5.12  161.17  167.82  174.48  1946 1.00
## y_rep[235]  161.02    0.11 5.08  154.63  161.07  167.68  2077 1.00
## y_rep[236]  150.06    0.11 4.97  143.81  150.11  156.18  2048 1.00
## y_rep[237]  153.79    0.12 5.09  147.26  153.82  160.34  1884 1.00
## y_rep[238]  163.24    0.13 5.16  156.64  163.17  169.65  1642 1.00
## y_rep[239]  154.77    0.11 5.10  148.40  154.63  161.46  2017 1.00
## y_rep[240]  151.55    0.12 5.24  144.96  151.47  158.03  1887 1.00
## y_rep[241]  151.82    0.13 5.21  145.17  151.86  158.42  1685 1.00
## y_rep[242]  152.32    0.12 5.09  145.84  152.46  158.71  1892 1.00
## y_rep[243]  152.83    0.13 5.12  146.24  152.80  159.56  1627 1.00
## y_rep[244]  150.97    0.12 5.18  144.19  151.04  157.45  1959 1.00
## y_rep[245]  152.71    0.12 4.97  146.50  152.65  158.92  1834 1.00
## y_rep[246]  155.76    0.12 5.10  148.99  155.81  162.31  1843 1.00
## y_rep[247]  157.26    0.12 5.14  150.64  157.35  163.61  1819 1.00
## y_rep[248]  152.47    0.11 5.14  145.96  152.49  159.03  2071 1.00
## y_rep[249]  157.73    0.11 5.00  151.40  157.74  164.08  1943 1.00
## y_rep[250]  155.42    0.11 5.07  149.04  155.46  161.94  2094 1.00
## y_rep[251]  161.78    0.11 5.18  155.14  161.85  168.31  2104 1.00
## y_rep[252]  157.03    0.12 5.25  150.23  157.06  163.62  2031 1.00
## y_rep[253]  153.18    0.12 5.25  146.57  153.14  159.93  1920 1.00
## y_rep[254]  153.33    0.12 5.07  146.89  153.40  159.83  1858 1.00
## y_rep[255]  162.05    0.12 5.20  155.52  162.04  168.69  1738 1.00
## y_rep[256]  154.28    0.12 5.09  147.86  154.31  160.73  1803 1.00
## y_rep[257]  156.56    0.11 5.10  150.09  156.58  163.11  2062 1.00
## y_rep[258]  142.44    0.12 5.19  135.97  142.41  149.15  1883 1.00
## y_rep[259]  165.06    0.12 5.23  158.56  164.97  171.71  1878 1.00
## y_rep[260]  164.50    0.12 5.10  158.02  164.54  170.88  1846 1.00
## y_rep[261]  156.92    0.11 5.12  150.61  156.95  163.46  1995 1.00
## y_rep[262]  154.90    0.12 5.14  148.16  155.04  161.35  1837 1.00
## y_rep[263]  155.07    0.12 5.22  148.31  155.01  161.70  1925 1.00
## y_rep[264]  158.27    0.13 5.09  151.82  158.22  164.90  1596 1.00
## y_rep[265]  155.92    0.11 5.09  149.43  156.01  162.29  1968 1.00
## y_rep[266]  153.69    0.11 5.00  147.11  153.86  160.06  1931 1.00
## y_rep[267]  164.85    0.12 5.08  158.25  164.86  171.54  1937 1.00
## y_rep[268]  150.20    0.12 5.11  143.72  150.17  156.70  1934 1.00
## y_rep[269]  159.91    0.11 5.12  153.38  159.92  166.52  2022 1.00
## y_rep[270]  155.71    0.12 5.26  148.93  155.65  162.30  1961 1.00
## y_rep[271]  158.59    0.12 5.15  152.08  158.52  165.13  1979 1.00
## y_rep[272]  153.70    0.11 5.10  147.04  153.75  160.27  1974 1.00
## y_rep[273]  159.89    0.11 5.09  153.16  159.92  166.49  2046 1.00
## y_rep[274]  150.74    0.12 5.32  144.19  150.65  157.50  1857 1.00
## y_rep[275]  147.71    0.12 5.14  141.26  147.51  154.15  1892 1.00
## y_rep[276]  160.44    0.11 5.28  153.63  160.58  167.12  2240 1.00
## y_rep[277]  154.37    0.11 5.08  147.95  154.38  161.02  2065 1.00
## y_rep[278]  153.27    0.11 5.10  146.71  153.42  159.74  1977 1.00
## y_rep[279]  156.21    0.13 5.22  149.36  156.34  162.69  1730 1.00
## y_rep[280]  149.75    0.12 5.10  143.08  149.79  156.29  1912 1.00
## y_rep[281]  145.28    0.12 5.25  138.45  145.34  151.91  1945 1.00
## y_rep[282]  149.39    0.12 5.10  142.52  149.52  155.71  1811 1.00
## y_rep[283]  147.78    0.12 5.10  141.12  147.87  154.29  1825 1.00
## y_rep[284]  157.84    0.12 5.07  151.50  157.84  164.32  1788 1.00
## y_rep[285]  146.06    0.12 5.16  139.37  146.12  152.64  1867 1.00
## y_rep[286]  153.66    0.11 4.96  147.50  153.75  160.09  1935 1.00
## y_rep[287]  157.48    0.12 5.30  150.71  157.44  164.01  1831 1.00
## y_rep[288]  159.89    0.11 5.15  153.31  159.85  166.48  2127 1.00
## y_rep[289]  153.78    0.12 5.08  147.11  153.95  160.19  1702 1.00
## y_rep[290]  157.33    0.11 5.07  151.10  157.25  163.81  2047 1.00
## y_rep[291]  156.10    0.12 5.14  149.51  156.18  162.61  1780 1.00
## y_rep[292]  165.15    0.12 5.23  158.44  165.30  171.81  2026 1.00
## y_rep[293]  157.86    0.12 5.16  151.11  157.84  164.35  1927 1.00
## y_rep[294]  157.47    0.11 5.23  150.69  157.51  164.09  2119 1.00
## y_rep[295]  156.93    0.11 5.17  150.41  156.95  163.47  2053 1.00
## y_rep[296]  145.90    0.12 5.17  139.46  145.89  152.56  1867 1.00
## y_rep[297]  147.04    0.12 5.09  140.49  147.01  153.48  1911 1.00
## y_rep[298]  158.16    0.12 5.19  151.54  158.11  164.77  1755 1.00
## y_rep[299]  160.06    0.12 5.10  153.47  160.05  166.55  1886 1.00
## y_rep[300]  155.99    0.12 5.23  149.33  156.11  162.72  1805 1.00
## y_rep[301]  147.46    0.12 5.12  140.87  147.44  154.16  1943 1.00
## y_rep[302]  157.82    0.11 5.07  151.43  157.75  164.37  1967 1.00
## y_rep[303]  148.35    0.12 5.16  141.84  148.41  154.80  1772 1.00
## y_rep[304]  156.63    0.11 5.05  149.91  156.80  163.02  2094 1.00
## y_rep[305]  164.00    0.12 5.19  157.09  164.21  170.65  1888 1.00
## y_rep[306]  162.85    0.12 5.15  156.46  162.83  169.47  1882 1.00
## y_rep[307]  157.76    0.11 5.08  151.27  157.81  164.32  2008 1.00
## y_rep[308]  153.07    0.11 5.07  146.56  153.16  159.50  2048 1.00
## y_rep[309]  144.91    0.12 5.27  137.93  144.90  151.66  2028 1.00
## y_rep[310]  159.16    0.12 5.06  152.98  159.12  165.72  1923 1.00
## y_rep[311]  154.01    0.12 5.15  147.31  154.12  160.40  1941 1.00
## y_rep[312]  144.26    0.12 5.26  137.44  144.35  150.97  1943 1.00
## y_rep[313]  156.55    0.12 5.14  150.01  156.54  163.18  1942 1.00
## y_rep[314]  155.78    0.12 5.20  149.21  155.82  162.38  1972 1.00
## y_rep[315]  156.71    0.12 5.09  150.12  156.54  163.52  1881 1.00
## y_rep[316]  150.95    0.12 5.06  144.60  150.94  157.37  1780 1.00
## y_rep[317]  159.43    0.11 5.12  152.60  159.51  165.99  2063 1.00
## y_rep[318]  154.88    0.11 5.13  148.18  154.80  161.44  2111 1.00
## y_rep[319]  152.19    0.12 5.14  145.74  152.08  158.73  1858 1.00
## y_rep[320]  151.56    0.12 5.07  145.13  151.45  158.20  1849 1.00
## y_rep[321]  154.85    0.12 5.06  148.40  154.81  161.12  1931 1.00
## y_rep[322]  160.24    0.12 5.26  153.38  160.31  167.00  1801 1.00
## y_rep[323]  158.47    0.11 5.00  152.29  158.44  164.63  2205 1.00
## y_rep[324]  160.48    0.12 5.11  154.06  160.42  166.93  1961 1.00
## y_rep[325]  153.74    0.12 5.04  147.21  153.59  160.28  1848 1.00
## y_rep[326]  146.01    0.12 5.19  139.36  145.97  152.72  1994 1.00
## y_rep[327]  152.55    0.12 5.28  145.79  152.51  159.33  1898 1.00
## y_rep[328]  156.04    0.12 5.18  149.43  156.13  162.64  1887 1.00
## y_rep[329]  151.81    0.11 5.10  145.34  151.79  158.36  2001 1.00
## y_rep[330]  148.39    0.12 5.11  142.08  148.35  154.87  1888 1.00
## y_rep[331]  153.12    0.11 5.09  146.54  153.16  159.56  1992 1.00
## y_rep[332]  162.52    0.11 5.14  155.96  162.34  169.07  2015 1.00
## y_rep[333]  152.68    0.12 5.18  146.20  152.49  159.48  2012 1.00
## y_rep[334]  149.86    0.12 5.24  143.08  149.80  156.72  2065 1.00
## y_rep[335]  153.08    0.11 5.15  146.66  153.09  159.80  2177 1.00
## y_rep[336]  161.73    0.12 5.17  155.06  161.81  168.52  2013 1.00
## y_rep[337]  148.72    0.11 5.19  141.93  148.71  155.50  2043 1.00
## y_rep[338]  152.55    0.12 5.15  145.86  152.47  159.30  1974 1.00
## y_rep[339]  146.42    0.12 5.12  140.00  146.37  153.13  1957 1.00
## y_rep[340]  153.75    0.12 5.10  147.24  153.76  160.24  1701 1.00
## y_rep[341]  148.36    0.12 5.16  141.65  148.42  154.88  1929 1.00
## y_rep[342]  153.73    0.11 5.06  147.15  153.60  160.46  2086 1.00
## y_rep[343]  153.93    0.12 5.16  147.34  154.05  160.24  1891 1.00
## y_rep[344]  157.05    0.12 5.16  150.74  156.88  163.76  1953 1.00
## y_rep[345]  149.43    0.13 5.10  142.96  149.39  155.93  1478 1.00
## y_rep[346]  151.09    0.12 5.09  144.49  151.19  157.41  1913 1.00
## y_rep[347]  150.99    0.12 5.19  144.33  151.10  157.49  1969 1.00
## y_rep[348]  156.47    0.12 5.19  149.84  156.32  163.32  2033 1.00
## y_rep[349]  144.67    0.12 5.15  138.25  144.59  151.43  1946 1.00
## y_rep[350]  161.04    0.12 5.27  154.58  161.01  167.98  2046 1.00
## y_rep[351]  162.79    0.12 5.02  156.18  162.83  169.26  1644 1.00
## y_rep[352]  161.62    0.11 5.00  155.19  161.75  167.96  2111 1.00
## lp__       -749.57    0.06 1.45 -751.51 -749.24 -748.08   564 1.00
## 
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:57:32 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).
summary(lm(height~weight.z+I(weight.z^2),data=howell2))
## 
## Call:
## lm(formula = height ~ weight.z + I(weight.z^2), data = howell2)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -19.7517  -2.8933   0.0313   3.1419  14.7936 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   154.60953    0.35209 439.120   <2e-16 ***
## weight.z        5.84515    0.27348  21.373   <2e-16 ***
## I(weight.z^2)  -0.01248    0.22483  -0.055    0.956    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.094 on 349 degrees of freedom
## Multiple R-squared:  0.5696, Adjusted R-squared:  0.5672 
## F-statistic:   231 on 2 and 349 DF,  p-value: < 2.2e-16

See Figure 4.9 of Statistical rethinking

https://github.com/ssp3nc3r/rethinking/blob/master/chapter04.Rmd

Here, all the variance is represented (variance of actual heights).

5.2 Howell 1: data set with all individuals

Standardizing the explanatory variable:

howell1 <- howell1 %>% mutate(weight.z = (weight - mean(weight)) / sd(weight))

Data in a list:

m7.data <- list(N = nrow(howell1), height = howell1$height, weight = howell1$weight.z)
data {                                                     // observed variables 
  int<lower=1> N;                                          // number of observations
  vector[N] height;                                        // response variables
  vector[N] weight;                                        // predictor (previously centered)
}

parameters {                                               // unobserved variables
  real beta[3];                                            // parameters: intercept and slope
  real sigma;
}

model {
  height ~ normal(beta[1] + beta[2]*weight + beta[3]*(weight .* weight),sigma);        // likelihood
  beta[1] ~ normal(178, 100);                              // prior of the intercept
  beta[2] ~ normal(0, 100);                                  // prior of the slope
  beta[3] ~ normal(0, 100);
  sigma ~ uniform(0,50);
}

generated quantities {
  vector[N] y_rep;

  for(i in 1:N)
  y_rep[i] = normal_rng(beta[1] + beta[2]*weight[i] + beta[3]*(weight[i] .* weight[i]), sigma);
}
fit.m7 <- sampling(m7, data = m7.data, iter = 1000, chains = 4, cores = 4)
print(fit.m7, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: c7fbeba14f23c52987bbbb1ee94a3a29.
## 4 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=2000.
## 
##                mean se_mean   sd      10%      50%      90% n_eff Rhat
## beta[1]      146.65    0.01 0.36   146.19   146.64   147.12   901 1.00
## beta[2]       21.42    0.01 0.27    21.06    21.42    21.76   597 1.01
## beta[3]       -8.40    0.01 0.28    -8.77    -8.40    -8.05   712 1.00
## sigma          5.77    0.00 0.18     5.55     5.77     6.00  2401 1.00
## y_rep[1]     158.78    0.13 5.73   151.57   158.81   165.97  1978 1.00
## y_rep[2]     148.02    0.13 5.79   140.74   148.02   155.29  1963 1.00
## y_rep[3]     140.64    0.13 5.92   132.97   140.75   148.15  1961 1.00
## y_rep[4]     160.20    0.13 5.68   152.71   160.24   167.62  2047 1.00
## y_rep[5]     153.52    0.14 5.81   146.05   153.51   161.07  1824 1.00
## y_rep[6]     157.46    0.14 5.93   149.74   157.42   165.19  1895 1.00
## y_rep[7]     150.15    0.13 5.82   142.68   150.33   157.44  1893 1.00
## y_rep[8]     160.32    0.12 5.65   153.27   160.21   167.48  2076 1.00
## y_rep[9]     145.63    0.13 5.65   138.16   145.71   152.86  1864 1.00
## y_rep[10]    160.21    0.13 5.61   152.95   160.16   167.56  1998 1.00
## y_rep[11]    159.64    0.13 5.70   152.26   159.59   166.98  2070 1.00
## y_rep[12]    153.49    0.13 5.83   146.06   153.42   161.07  1994 1.00
## y_rep[13]    147.28    0.13 5.76   140.05   147.25   154.60  1968 1.00
## y_rep[14]    158.59    0.13 5.77   151.28   158.67   166.10  2035 1.00
## y_rep[15]    143.92    0.13 5.87   136.10   144.00   151.52  2123 1.00
## y_rep[16]    159.01    0.13 5.76   151.62   159.16   166.42  1859 1.00
## y_rep[17]    154.47    0.13 5.73   146.96   154.41   161.77  1963 1.00
## y_rep[18]    150.45    0.14 5.88   142.95   150.52   158.11  1895 1.00
## y_rep[19]    113.36    0.13 5.80   106.08   113.26   120.78  2021 1.00
## y_rep[20]     97.13    0.13 5.69    89.84    97.06   104.28  1909 1.00
## y_rep[21]     85.82    0.14 5.84    78.51    85.82    93.26  1764 1.00
## y_rep[22]    159.14    0.13 5.79   151.87   159.07   166.45  1953 1.00
## y_rep[23]    154.85    0.12 5.69   147.69   154.86   162.04  2209 1.00
## y_rep[24]    123.54    0.13 5.86   115.90   123.55   131.15  1939 1.00
## y_rep[25]    103.07    0.13 5.85    95.76   102.94   110.63  1940 1.00
## y_rep[26]    146.68    0.13 5.83   139.39   146.73   154.21  1974 1.00
## y_rep[27]    149.77    0.13 5.72   142.39   149.93   157.14  1906 1.00
## y_rep[28]    146.43    0.15 5.74   139.19   146.42   153.73  1467 1.00
## y_rep[29]    132.13    0.13 5.72   124.58   131.99   139.65  1885 1.00
## y_rep[30]    121.27    0.13 5.69   113.88   121.09   128.85  2004 1.00
## y_rep[31]    108.58    0.14 6.01   100.86   108.55   116.59  1832 1.00
## y_rep[32]    152.70    0.13 5.78   145.23   152.76   160.18  1956 1.00
## y_rep[33]    160.13    0.14 5.90   152.56   160.08   167.75  1910 1.00
## y_rep[34]    149.10    0.13 5.90   141.52   149.12   156.69  2081 1.00
## y_rep[35]    150.44    0.13 5.71   143.18   150.50   157.77  1825 1.00
## y_rep[36]    157.98    0.13 5.71   150.48   158.06   165.33  2063 1.00
## y_rep[37]    150.80    0.13 5.71   143.32   150.87   157.98  1990 1.00
## y_rep[38]    146.49    0.12 5.62   139.17   146.54   153.76  2122 1.00
## y_rep[39]    158.26    0.12 5.58   151.17   158.17   165.51  2208 1.00
## y_rep[40]    158.59    0.14 5.90   150.82   158.59   166.19  1658 1.00
## y_rep[41]    159.19    0.13 5.84   151.85   159.08   166.86  2033 1.00
## y_rep[42]    159.30    0.13 5.85   152.05   159.14   166.92  1955 1.00
## y_rep[43]    159.94    0.14 5.84   152.36   160.13   167.27  1666 1.00
## y_rep[44]    151.32    0.14 5.79   143.76   151.42   158.56  1792 1.00
## y_rep[45]    159.50    0.13 5.84   151.84   159.41   166.91  1932 1.00
## y_rep[46]    134.91    0.14 5.73   127.64   134.80   142.25  1714 1.00
## y_rep[47]    115.53    0.14 5.88   107.94   115.45   123.09  1675 1.00
## y_rep[48]    123.05    0.12 5.69   116.07   122.80   130.57  2220 1.00
## y_rep[49]     94.71    0.13 5.67    87.59    94.79   101.95  1891 1.00
## y_rep[50]    153.53    0.13 5.88   146.04   153.63   160.99  1898 1.00
## y_rep[51]    150.57    0.12 5.68   142.99   150.59   157.89  2156 1.00
## y_rep[52]    155.16    0.13 5.75   147.73   155.23   162.57  2107 1.00
## y_rep[53]    156.76    0.12 5.76   149.39   156.76   163.96  2127 1.00
## y_rep[54]    119.70    0.13 5.81   112.29   119.67   127.17  1869 1.00
## y_rep[55]    101.51    0.13 5.75    94.36   101.55   108.71  1898 1.00
## y_rep[56]     93.14    0.14 5.85    85.88    92.96   100.97  1739 1.00
## y_rep[57]    159.52    0.13 5.80   152.12   159.55   166.88  2096 1.00
## y_rep[58]    157.98    0.13 5.74   150.60   158.03   165.43  1881 1.00
## y_rep[59]    156.39    0.13 6.03   148.74   156.42   164.20  2000 1.00
## y_rep[60]    159.88    0.14 5.85   152.43   159.81   167.55  1798 1.00
## y_rep[61]    160.26    0.13 5.82   152.68   160.41   167.81  1977 1.00
## y_rep[62]    157.50    0.13 5.81   150.12   157.52   164.88  2147 1.00
## y_rep[63]    158.78    0.13 5.64   151.62   158.81   165.96  1938 1.00
## y_rep[64]    160.09    0.13 5.86   152.57   160.09   167.56  1953 1.00
## y_rep[65]    158.60    0.14 5.87   151.08   158.53   166.31  1655 1.00
## y_rep[66]    157.30    0.13 5.76   149.83   157.26   164.64  2004 1.00
## y_rep[67]    154.73    0.13 5.73   147.41   154.74   162.00  1908 1.00
## y_rep[68]    147.92    0.13 5.54   140.88   147.93   155.04  1836 1.00
## y_rep[69]    160.15    0.13 5.89   152.72   160.03   167.57  1993 1.00
## y_rep[70]    148.64    0.13 5.89   140.93   148.77   155.95  2129 1.01
## y_rep[71]    125.74    0.13 5.81   118.20   125.83   133.26  1944 1.00
## y_rep[72]    128.25    0.13 5.68   120.82   128.31   135.51  2009 1.00
## y_rep[73]    159.10    0.13 5.67   151.91   159.05   166.39  1884 1.00
## y_rep[74]    148.92    0.13 5.73   141.44   149.01   156.25  1969 1.00
## y_rep[75]    136.57    0.13 5.89   128.85   136.76   144.09  1973 1.00
## y_rep[76]    158.33    0.13 5.81   150.93   158.20   165.71  2127 1.00
## y_rep[77]    158.29    0.13 5.74   150.93   158.23   165.73  2082 1.00
## y_rep[78]     93.78    0.13 5.55    86.65    93.73   100.95  1826 1.00
## y_rep[79]    160.02    0.13 5.77   152.63   160.05   167.34  1861 1.00
## y_rep[80]    149.81    0.14 5.86   142.20   149.83   157.30  1825 1.00
## y_rep[81]    154.21    0.13 5.71   146.77   154.28   161.70  1873 1.00
## y_rep[82]    132.38    0.13 5.70   125.07   132.44   139.57  1941 1.00
## y_rep[83]    158.36    0.13 5.72   151.10   158.18   165.77  1976 1.00
## y_rep[84]    155.65    0.13 5.71   148.27   155.67   162.90  2015 1.00
## y_rep[85]    159.66    0.13 5.69   152.40   159.85   166.85  1908 1.00
## y_rep[86]    151.48    0.14 5.94   143.92   151.41   159.03  1908 1.00
## y_rep[87]    107.45    0.13 5.94    99.80   107.41   114.98  2179 1.00
## y_rep[88]    159.87    0.13 5.83   152.30   159.81   167.35  1980 1.00
## y_rep[89]    151.63    0.13 5.83   144.17   151.68   158.83  1996 1.00
## y_rep[90]    147.51    0.14 5.89   139.89   147.57   154.91  1890 1.00
## y_rep[91]    157.89    0.14 5.77   150.29   158.05   165.22  1760 1.00
## y_rep[92]    154.74    0.13 5.77   147.31   154.75   161.82  2015 1.00
## y_rep[93]    158.56    0.13 5.88   150.97   158.53   166.24  2008 1.00
## y_rep[94]    160.27    0.13 5.74   152.78   160.35   167.40  1852 1.00
## y_rep[95]    111.24    0.13 5.83   103.59   111.22   118.70  2083 1.00
## y_rep[96]    158.93    0.13 5.82   151.41   159.07   166.32  2157 1.00
## y_rep[97]    155.14    0.13 5.76   147.50   155.16   162.68  1941 1.00
## y_rep[98]    145.92    0.13 5.98   138.14   145.80   153.54  2069 1.00
## y_rep[99]    108.88    0.13 5.93   101.10   108.88   116.48  2120 1.00
## y_rep[100]   160.00    0.14 5.75   152.89   159.80   167.50  1674 1.00
## y_rep[101]   132.25    0.13 5.90   124.58   132.17   139.77  1927 1.00
## y_rep[102]   159.91    0.13 5.82   152.53   160.00   167.34  2141 1.00
## y_rep[103]   158.32    0.14 5.95   151.10   158.14   166.02  1919 1.00
## y_rep[104]   153.39    0.13 5.96   145.77   153.37   161.02  2099 1.00
## y_rep[105]   141.92    0.13 5.80   134.43   142.01   149.44  1978 1.00
## y_rep[106]   160.14    0.14 5.88   152.63   160.06   167.75  1870 1.00
## y_rep[107]   155.55    0.13 5.66   148.37   155.52   162.67  2008 1.00
## y_rep[108]   152.31    0.13 5.85   144.82   152.41   159.60  2045 1.00
## y_rep[109]   141.61    0.12 5.74   134.42   141.56   149.02  2201 1.00
## y_rep[110]   106.44    0.15 5.83    99.16   106.40   113.88  1443 1.00
## y_rep[111]   104.74    0.13 5.98    97.22   104.62   112.49  1983 1.00
## y_rep[112]   157.36    0.14 5.77   149.95   157.28   164.62  1677 1.00
## y_rep[113]   153.48    0.13 5.79   146.21   153.56   160.65  1953 1.00
## y_rep[114]   109.67    0.13 5.75   102.10   109.75   117.06  1884 1.00
## y_rep[115]   100.59    0.13 5.73    93.19   100.49   107.79  2038 1.00
## y_rep[116]    88.95    0.13 5.97    81.38    88.89    96.53  1975 1.00
## y_rep[117]   158.36    0.14 5.78   150.97   158.42   165.57  1826 1.00
## y_rep[118]   149.15    0.14 5.96   141.55   149.15   156.85  1895 1.00
## y_rep[119]   156.77    0.13 5.72   149.34   156.77   163.98  1839 1.00
## y_rep[120]   154.84    0.12 5.76   147.41   154.66   162.29  2156 1.00
## y_rep[121]   160.36    0.14 5.98   152.64   160.26   168.07  1916 1.00
## y_rep[122]   160.29    0.14 5.80   152.75   160.16   167.73  1812 1.00
## y_rep[123]   160.05    0.14 5.80   152.68   160.07   167.45  1753 1.00
## y_rep[124]    83.29    0.13 5.80    76.02    83.20    90.77  1998 1.00
## y_rep[125]   154.74    0.13 5.81   147.31   154.86   162.01  2130 1.00
## y_rep[126]   159.68    0.13 5.83   152.04   159.53   167.48  2057 1.00
## y_rep[127]    94.34    0.13 5.84    86.95    94.43   101.73  1959 1.00
## y_rep[128]   153.88    0.14 5.94   146.24   153.95   161.54  1719 1.00
## y_rep[129]   160.00    0.13 5.83   152.73   159.80   167.62  1924 1.00
## y_rep[130]   154.46    0.13 5.99   146.61   154.36   161.97  2098 1.00
## y_rep[131]   112.89    0.13 5.88   105.45   112.69   120.58  1899 1.00
## y_rep[132]   100.40    0.13 5.95    93.05   100.24   108.02  2073 1.00
## y_rep[133]   160.14    0.13 5.83   152.38   160.17   167.78  1997 1.00
## y_rep[134]   159.72    0.13 5.79   152.29   159.79   167.01  1910 1.00
## y_rep[135]   154.96    0.14 5.80   147.64   154.90   162.40  1788 1.00
## y_rep[136]   159.26    0.13 5.84   151.76   159.31   166.80  1915 1.00
## y_rep[137]   150.71    0.13 5.63   143.43   150.61   157.88  1807 1.00
## y_rep[138]   159.54    0.13 5.77   152.25   159.46   166.93  2019 1.00
## y_rep[139]   135.98    0.13 5.67   128.62   136.04   143.18  1986 1.00
## y_rep[140]   159.16    0.13 6.00   151.40   159.17   166.82  2076 1.00
## y_rep[141]    89.61    0.13 5.74    82.30    89.52    97.23  1899 1.00
## y_rep[142]   158.84    0.13 5.90   151.40   158.81   166.51  2004 1.00
## y_rep[143]   151.48    0.13 5.79   144.29   151.49   159.00  2007 1.00
## y_rep[144]   150.24    0.13 5.67   142.76   150.18   157.50  1946 1.00
## y_rep[145]   119.78    0.13 5.84   112.64   119.67   127.19  2058 1.00
## y_rep[146]   148.49    0.13 5.80   140.96   148.50   155.78  1904 1.00
## y_rep[147]   154.66    0.13 5.80   147.18   154.82   161.98  1880 1.00
## y_rep[148]   113.98    0.13 5.79   106.49   114.06   121.26  1955 1.00
## y_rep[149]   105.47    0.13 5.78    98.21   105.28   113.13  1959 1.00
## y_rep[150]   156.60    0.13 5.83   149.21   156.54   164.17  1907 1.00
## y_rep[151]   158.66    0.13 5.62   151.40   158.70   165.63  1958 1.00
## y_rep[152]   117.42    0.14 5.87   109.80   117.29   124.84  1767 1.00
## y_rep[153]   111.78    0.14 5.79   104.54   111.71   119.27  1777 1.00
## y_rep[154]   160.33    0.13 5.75   152.79   160.46   167.55  1936 1.00
## y_rep[155]   142.49    0.13 5.81   135.30   142.26   150.18  2029 1.00
## y_rep[156]   153.34    0.13 5.83   146.00   153.34   160.89  1926 1.00
## y_rep[157]    94.96    0.14 5.88    87.45    94.88   102.52  1828 1.00
## y_rep[158]   159.67    0.13 5.67   152.31   159.68   166.99  1979 1.00
## y_rep[159]   159.11    0.13 5.70   151.83   158.94   166.45  1963 1.00
## y_rep[160]   160.38    0.13 5.80   153.27   160.18   167.82  1886 1.00
## y_rep[161]   160.27    0.13 5.78   152.76   160.19   167.73  1971 1.00
## y_rep[162]   160.37    0.14 5.85   153.02   160.32   167.96  1786 1.00
## y_rep[163]   154.59    0.13 5.89   147.32   154.55   162.05  1947 1.00
## y_rep[164]   120.46    0.13 5.82   113.08   120.44   127.97  2078 1.00
## y_rep[165]   153.20    0.13 5.78   145.59   153.31   160.32  2077 1.00
## y_rep[166]   159.51    0.13 5.77   152.04   159.60   166.65  2011 1.00
## y_rep[167]   156.29    0.13 5.71   148.94   156.21   163.67  2002 1.00
## y_rep[168]   157.38    0.13 5.75   150.04   157.21   164.72  1937 1.00
## y_rep[169]   154.40    0.13 5.81   147.13   154.35   161.75  1882 1.00
## y_rep[170]   159.93    0.13 5.71   152.59   159.99   167.30  2024 1.00
## y_rep[171]    93.16    0.14 5.86    85.96    93.04   100.96  1784 1.00
## y_rep[172]    93.90    0.13 5.83    86.33    93.89   101.29  2161 1.00
## y_rep[173]   156.20    0.13 5.66   148.91   156.19   163.44  1934 1.00
## y_rep[174]   141.43    0.13 5.85   134.07   141.42   149.14  1985 1.00
## y_rep[175]   159.89    0.13 5.81   152.38   159.82   167.30  2053 1.00
## y_rep[176]   160.19    0.13 5.84   152.63   160.13   167.83  1907 1.00
## y_rep[177]   147.77    0.13 5.89   140.21   147.81   155.31  1916 1.00
## y_rep[178]   159.07    0.14 5.90   151.41   158.84   166.62  1865 1.00
## y_rep[179]   160.13    0.15 5.92   152.44   160.10   167.78  1627 1.00
## y_rep[180]   154.30    0.13 5.77   146.87   154.37   161.76  1990 1.00
## y_rep[181]   159.59    0.13 5.80   152.01   159.63   167.15  1997 1.00
## y_rep[182]   157.86    0.13 5.82   150.51   157.96   165.49  1922 1.00
## y_rep[183]   158.35    0.14 5.90   150.76   158.46   165.89  1823 1.00
## y_rep[184]   159.17    0.14 5.82   151.50   159.20   166.87  1721 1.00
## y_rep[185]   140.16    0.12 5.94   132.72   140.06   147.74  2271 1.00
## y_rep[186]   141.20    0.13 5.70   133.88   141.32   148.45  1785 1.00
## y_rep[187]   123.82    0.13 5.80   116.29   123.85   131.33  2035 1.00
## y_rep[188]   125.36    0.14 5.67   118.02   125.24   132.51  1744 1.00
## y_rep[189]   118.57    0.13 5.87   111.17   118.42   126.17  2018 1.00
## y_rep[190]   146.27    0.13 5.85   138.56   146.36   153.60  1931 1.00
## y_rep[191]   157.37    0.13 5.80   149.80   157.44   164.91  1950 1.00
## y_rep[192]   156.01    0.13 5.75   148.68   155.98   163.36  1925 1.00
## y_rep[193]   157.29    0.13 5.64   150.00   157.22   164.50  1946 1.00
## y_rep[194]   151.24    0.13 5.84   143.64   151.21   158.88  1997 1.00
## y_rep[195]   153.88    0.15 5.88   145.98   153.90   161.22  1626 1.00
## y_rep[196]    94.28    0.14 5.85    86.91    94.19   101.80  1851 1.00
## y_rep[197]   157.08    0.13 5.70   149.56   157.13   164.40  1941 1.00
## y_rep[198]   160.28    0.13 5.71   153.25   160.11   167.72  1969 1.00
## y_rep[199]   160.17    0.13 5.71   152.72   160.34   167.49  1953 1.00
## y_rep[200]   110.11    0.13 5.67   102.75   110.07   117.57  2031 1.00
## y_rep[201]    96.19    0.13 5.84    88.72    96.15   103.64  2113 1.00
## y_rep[202]   151.23    0.13 5.83   143.80   151.18   158.87  1944 1.00
## y_rep[203]    92.26    0.13 5.84    84.44    92.45    99.49  1973 1.00
## y_rep[204]   157.62    0.12 5.74   150.28   157.55   164.97  2158 1.00
## y_rep[205]   152.72    0.13 5.75   145.37   152.77   160.02  1872 1.00
## y_rep[206]   160.29    0.14 5.96   152.66   160.48   167.85  1942 1.01
## y_rep[207]   150.81    0.13 5.63   143.67   150.94   158.09  1814 1.00
## y_rep[208]   151.34    0.13 5.75   143.83   151.33   158.64  2089 1.00
## y_rep[209]   156.41    0.13 5.76   149.13   156.27   163.79  2032 1.00
## y_rep[210]   151.68    0.14 5.81   143.99   151.80   158.97  1806 1.00
## y_rep[211]   139.26    0.14 5.88   131.89   139.18   146.70  1732 1.00
## y_rep[212]   157.87    0.13 5.77   150.50   157.83   165.25  1945 1.00
## y_rep[213]   120.92    0.13 5.66   113.63   120.88   128.20  1957 1.00
## y_rep[214]   152.93    0.14 5.88   145.58   152.89   160.68  1867 1.00
## y_rep[215]   159.70    0.13 5.91   152.13   159.57   167.47  1977 1.00
## y_rep[216]   153.39    0.14 5.92   145.91   153.28   161.09  1856 1.00
## y_rep[217]   107.62    0.13 5.85   100.51   107.57   114.92  1928 1.00
## y_rep[218]   160.30    0.14 5.81   152.73   160.44   167.68  1789 1.00
## y_rep[219]   156.89    0.13 5.77   149.58   156.89   164.45  1989 1.00
## y_rep[220]   156.62    0.13 5.78   149.28   156.72   164.08  2008 1.00
## y_rep[221]    98.35    0.14 5.86    90.79    98.40   105.78  1874 1.00
## y_rep[222]   111.86    0.13 5.76   104.57   111.93   119.13  1922 1.00
## y_rep[223]   147.15    0.12 5.74   139.86   147.20   154.26  2154 1.00
## y_rep[224]   157.18    0.13 5.67   149.83   157.09   164.52  2037 1.00
## y_rep[225]   158.86    0.13 5.76   151.33   158.86   166.09  1944 1.00
## y_rep[226]   157.33    0.13 5.70   149.92   157.35   164.70  1962 1.00
## y_rep[227]   116.58    0.14 5.82   109.05   116.82   123.91  1637 1.00
## y_rep[228]   158.97    0.14 5.86   151.29   159.19   166.18  1764 1.00
## y_rep[229]   153.60    0.13 5.78   146.34   153.56   161.13  2028 1.00
## y_rep[230]   150.26    0.13 5.78   142.87   150.24   157.55  1958 1.00
## y_rep[231]   154.37    0.13 5.90   146.54   154.38   161.99  1911 1.00
## y_rep[232]   107.92    0.13 5.89   100.52   108.09   115.33  2042 1.00
## y_rep[233]   150.41    0.13 5.87   142.98   150.28   158.02  2205 1.00
## y_rep[234]   159.22    0.13 5.66   152.05   159.18   166.67  2001 1.00
## y_rep[235]   150.82    0.13 5.76   143.33   150.82   158.22  1861 1.00
## y_rep[236]   152.25    0.14 5.79   144.92   152.27   159.74  1752 1.00
## y_rep[237]   159.68    0.14 6.06   152.12   159.57   167.51  1876 1.00
## y_rep[238]   160.41    0.13 5.70   153.23   160.43   167.61  2073 1.00
## y_rep[239]   157.53    0.12 5.65   150.14   157.62   164.76  2112 1.00
## y_rep[240]   152.90    0.13 5.71   145.79   152.97   160.06  1882 1.00
## y_rep[241]   131.52    0.12 5.57   124.36   131.52   138.59  2060 1.00
## y_rep[242]   151.54    0.13 5.76   144.14   151.69   158.91  2104 1.00
## y_rep[243]   115.83    0.13 5.63   108.65   115.72   123.02  1825 1.00
## y_rep[244]   155.43    0.13 5.78   147.80   155.57   162.57  1995 1.00
## y_rep[245]   140.75    0.13 5.61   133.72   140.76   148.40  1850 1.00
## y_rep[246]   116.75    0.13 5.73   109.42   116.82   123.87  2100 1.00
## y_rep[247]   157.29    0.13 5.78   149.91   157.35   164.83  1925 1.00
## y_rep[248]   149.87    0.13 5.81   142.45   149.82   157.25  1947 1.00
## y_rep[249]   103.10    0.14 5.84    95.90   103.11   110.52  1863 1.00
## y_rep[250]   147.37    0.12 5.69   139.92   147.27   154.62  2083 1.00
## y_rep[251]   153.26    0.13 5.67   145.84   153.41   160.06  1868 1.00
## y_rep[252]   158.99    0.13 5.71   151.96   158.85   166.40  2039 1.00
## y_rep[253]   158.50    0.14 5.88   150.69   158.54   165.88  1727 1.00
## y_rep[254]   157.59    0.13 5.80   150.09   157.73   164.69  2036 1.00
## y_rep[255]   153.60    0.13 5.94   145.84   153.73   161.23  2115 1.00
## y_rep[256]   147.91    0.13 5.74   140.55   148.08   155.24  1937 1.00
## y_rep[257]   155.41    0.13 5.69   148.15   155.55   162.79  2002 1.00
## y_rep[258]   159.11    0.13 5.72   151.84   159.12   166.35  2069 1.00
## y_rep[259]   148.23    0.13 5.62   141.00   148.16   155.45  1867 1.00
## y_rep[260]   160.10    0.13 6.00   152.72   160.00   167.37  2032 1.00
## y_rep[261]   150.75    0.13 5.75   143.65   150.69   158.21  1827 1.00
## y_rep[262]   158.56    0.13 5.80   151.12   158.60   166.14  1849 1.00
## y_rep[263]   147.38    0.14 5.85   139.86   147.47   154.80  1787 1.00
## y_rep[264]    97.98    0.14 5.90    90.55    98.01   105.28  1789 1.00
## y_rep[265]   153.64    0.13 5.74   146.42   153.69   161.04  2000 1.00
## y_rep[266]   100.83    0.14 5.87    93.43   100.78   108.35  1750 1.00
## y_rep[267]   156.53    0.14 5.85   149.12   156.48   164.14  1771 1.00
## y_rep[268]   159.89    0.13 5.64   152.58   159.79   166.94  1753 1.00
## y_rep[269]   160.19    0.13 5.80   152.61   160.32   167.65  1932 1.00
## y_rep[270]   149.92    0.13 5.69   142.48   149.89   157.07  1986 1.00
## y_rep[271]   146.39    0.14 5.81   139.18   146.28   154.02  1848 1.00
## y_rep[272]   159.82    0.14 6.00   152.14   159.68   167.80  1865 1.00
## y_rep[273]   118.22    0.12 5.68   111.02   118.31   125.45  2114 1.00
## y_rep[274]   156.41    0.14 5.74   149.11   156.30   163.88  1593 1.00
## y_rep[275]   156.63    0.13 5.79   149.40   156.76   163.86  1935 1.00
## y_rep[276]   156.60    0.12 5.70   149.43   156.58   163.95  2163 1.00
## y_rep[277]   156.64    0.13 5.82   149.34   156.67   164.20  1963 1.00
## y_rep[278]   152.88    0.13 5.82   145.41   152.67   160.24  1955 1.00
## y_rep[279]   156.25    0.13 5.80   148.67   156.13   163.68  2001 1.00
## y_rep[280]   125.78    0.14 5.82   118.30   125.73   133.37  1854 1.00
## y_rep[281]   144.81    0.14 5.80   137.53   144.86   152.13  1824 1.00
## y_rep[282]   142.08    0.14 5.95   134.49   142.10   149.71  1929 1.00
## y_rep[283]   147.11    0.14 5.85   139.65   147.08   154.51  1858 1.00
## y_rep[284]   159.46    0.13 5.75   152.17   159.56   166.77  2014 1.00
## y_rep[285]   160.44    0.13 5.71   153.22   160.40   167.76  2001 1.00
## y_rep[286]   148.76    0.13 5.78   141.42   148.78   155.99  1969 1.00
## y_rep[287]   152.92    0.13 5.62   145.81   152.72   160.40  1921 1.00
## y_rep[288]   159.81    0.14 5.89   152.19   159.88   167.34  1755 1.00
## y_rep[289]   156.03    0.13 5.77   148.59   156.08   163.39  1981 1.00
## y_rep[290]   143.85    0.13 5.83   136.11   144.00   151.45  2048 1.00
## y_rep[291]   117.94    0.13 5.69   110.86   117.79   125.52  1962 1.00
## y_rep[292]   105.04    0.13 5.85    97.43   105.19   112.58  2180 1.00
## y_rep[293]   159.56    0.13 5.72   152.13   159.68   166.95  2005 1.00
## y_rep[294]   143.53    0.13 5.90   136.17   143.54   151.08  1994 1.00
## y_rep[295]   155.85    0.14 5.75   148.56   156.03   163.15  1808 1.00
## y_rep[296]   158.69    0.13 5.58   151.49   158.75   165.74  1943 1.00
## y_rep[297]   155.01    0.13 5.68   147.67   154.98   162.17  1824 1.00
## y_rep[298]   145.48    0.13 5.65   138.39   145.53   152.76  1966 1.00
## y_rep[299]   124.55    0.13 5.75   117.28   124.40   131.87  2006 1.00
## y_rep[300]   144.21    0.13 5.78   136.87   144.21   151.55  2014 1.00
## y_rep[301]    90.72    0.13 5.73    83.32    90.90    97.99  1845 1.00
## y_rep[302]   153.61    0.13 5.77   146.30   153.58   161.13  1854 1.00
## y_rep[303]   151.76    0.13 5.86   144.31   151.82   159.22  2045 1.00
## y_rep[304]   147.06    0.14 6.03   139.55   147.12   154.82  1924 1.00
## y_rep[305]   159.47    0.13 5.69   152.11   159.59   166.66  1990 1.00
## y_rep[306]   157.58    0.13 5.86   150.12   157.53   165.00  1910 1.00
## y_rep[307]   127.39    0.13 5.85   119.83   127.59   134.90  1962 1.00
## y_rep[308]   101.09    0.13 5.69    93.83   100.97   108.44  2050 1.00
## y_rep[309]   159.56    0.14 5.71   152.20   159.61   166.95  1628 1.00
## y_rep[310]   159.46    0.13 5.88   151.82   159.26   166.96  2015 1.00
## y_rep[311]   155.74    0.12 5.65   148.67   155.71   162.90  2169 1.00
## y_rep[312]   108.70    0.14 6.06   100.96   108.85   116.25  1985 1.00
## y_rep[313]   160.12    0.13 5.70   152.94   160.15   167.27  1936 1.00
## y_rep[314]   159.70    0.13 5.79   152.27   159.72   167.09  2047 1.00
## y_rep[315]   144.57    0.14 5.83   137.02   144.55   152.04  1724 1.00
## y_rep[316]   151.78    0.13 5.72   144.47   151.78   159.00  1997 1.00
## y_rep[317]   159.32    0.13 5.81   151.89   159.33   166.90  1982 1.00
## y_rep[318]   152.49    0.13 5.86   144.96   152.43   159.99  1988 1.00
## y_rep[319]   159.98    0.13 5.87   152.34   160.14   167.45  2036 1.00
## y_rep[320]   152.08    0.13 5.88   144.50   152.33   159.39  2096 1.00
## y_rep[321]   156.12    0.13 5.77   148.69   156.18   163.25  2071 1.00
## y_rep[322]   160.30    0.13 5.79   152.88   160.29   167.56  1892 1.00
## y_rep[323]   156.90    0.13 5.82   149.44   156.77   164.07  1988 1.00
## y_rep[324]    88.84    0.13 5.69    81.43    88.88    96.16  1979 1.00
## y_rep[325]    89.59    0.13 5.80    82.17    89.61    97.11  1922 1.00
## y_rep[326]    69.30    0.13 5.71    61.87    69.39    76.67  1928 1.00
## y_rep[327]    90.98    0.13 5.89    83.49    90.95    98.40  1908 1.00
## y_rep[328]   109.91    0.13 5.58   102.82   109.93   116.87  1952 1.00
## y_rep[329]    88.82    0.13 5.73    81.79    88.80    96.22  1994 1.00
## y_rep[330]   153.64    0.12 5.68   146.12   153.67   161.06  2079 1.00
## y_rep[331]   104.10    0.13 5.97    96.37   104.25   111.83  1974 1.00
## y_rep[332]    72.46    0.14 5.94    64.81    72.37    80.15  1888 1.00
## y_rep[333]    90.20    0.13 5.72    82.78    90.28    97.73  1883 1.00
## y_rep[334]   100.11    0.13 5.71    92.52   100.14   107.31  2010 1.00
## y_rep[335]    83.51    0.13 5.91    76.02    83.53    91.08  1950 1.00
## y_rep[336]   154.37    0.14 5.76   146.89   154.39   161.80  1816 1.00
## y_rep[337]   154.69    0.14 5.76   147.05   154.82   162.01  1709 1.00
## y_rep[338]   135.31    0.13 5.69   128.02   135.39   142.65  1913 1.00
## y_rep[339]    93.49    0.13 5.90    85.88    93.45   101.31  1988 1.00
## y_rep[340]   155.48    0.13 5.88   147.78   155.57   163.02  1999 1.00
## y_rep[341]   153.36    0.13 5.88   146.08   153.35   160.77  2161 1.00
## y_rep[342]   155.35    0.13 5.90   147.60   155.47   162.80  2083 1.00
## y_rep[343]   157.80    0.12 5.78   150.19   157.72   165.39  2245 1.00
## y_rep[344]   115.87    0.13 5.73   108.45   115.99   123.11  1825 1.00
## y_rep[345]   102.70    0.13 5.69    95.48   102.80   110.15  1863 1.00
## y_rep[346]   158.68    0.14 5.87   151.06   158.72   166.08  1836 1.00
## y_rep[347]   154.77    0.13 5.74   147.34   154.74   162.30  1840 1.00
## y_rep[348]   106.19    0.13 5.81    98.63   106.18   113.63  1943 1.00
## y_rep[349]    94.47    0.14 5.77    87.03    94.53   101.72  1600 1.00
## y_rep[350]   159.09    0.13 5.81   151.54   159.03   166.40  1903 1.00
## y_rep[351]   157.63    0.13 5.77   150.38   157.56   164.85  2011 1.00
## y_rep[352]   160.26    0.13 5.91   152.89   160.05   167.96  2090 1.00
## y_rep[353]   158.71    0.13 5.70   151.17   158.68   165.99  1891 1.00
## y_rep[354]    93.77    0.13 5.64    86.67    93.74   101.23  1884 1.00
## y_rep[355]    68.10    0.13 5.84    60.64    68.22    75.44  2007 1.00
## y_rep[356]   135.10    0.13 5.84   127.61   135.05   142.52  2079 1.00
## y_rep[357]   155.82    0.14 5.73   148.72   155.87   162.99  1712 1.00
## y_rep[358]   155.54    0.13 5.69   148.39   155.52   162.75  1930 1.00
## y_rep[359]    88.85    0.13 5.74    81.31    88.93    96.12  1913 1.00
## y_rep[360]    89.72    0.13 5.72    82.44    89.74    97.06  2051 1.00
## y_rep[361]    75.08    0.14 5.81    67.39    75.07    82.35  1782 1.00
## y_rep[362]    92.90    0.13 5.65    85.39    92.83   100.01  1908 1.00
## y_rep[363]    73.97    0.12 5.67    66.63    74.00    81.36  2107 1.00
## y_rep[364]    91.66    0.13 5.89    84.25    91.65    99.18  2049 1.00
## y_rep[365]    76.08    0.13 5.98    68.36    76.29    83.76  1991 1.00
## y_rep[366]   160.15    0.14 5.76   152.87   160.26   167.41  1745 1.00
## y_rep[367]   156.84    0.13 5.85   149.47   157.05   164.27  2018 1.00
## y_rep[368]    92.22    0.12 5.70    84.75    92.14    99.57  2137 1.00
## y_rep[369]    93.07    0.14 5.83    85.56    92.85   100.74  1758 1.00
## y_rep[370]    74.59    0.13 5.80    67.18    74.59    82.07  1944 1.00
## y_rep[371]    81.03    0.13 5.71    73.79    80.91    88.56  2017 1.00
## y_rep[372]   158.43    0.13 5.80   150.82   158.39   165.95  2023 1.00
## y_rep[373]   134.80    0.13 5.86   127.18   134.68   142.41  1928 1.00
## y_rep[374]   140.08    0.14 5.98   132.58   140.16   147.61  1773 1.00
## y_rep[375]   160.11    0.13 5.70   152.80   160.09   167.48  1963 1.00
## y_rep[376]   114.49    0.13 5.82   107.05   114.44   122.12  1970 1.00
## y_rep[377]    78.68    0.13 5.71    71.13    78.78    85.86  1952 1.00
## y_rep[378]    78.26    0.14 5.78    70.87    78.32    85.76  1780 1.00
## y_rep[379]    89.52    0.13 5.84    82.06    89.44    97.05  2079 1.00
## y_rep[380]   160.22    0.13 5.93   152.54   160.41   167.72  2115 1.00
## y_rep[381]    76.94    0.12 5.53    70.09    76.87    84.26  1958 1.00
## y_rep[382]   131.74    0.13 5.79   124.32   131.89   138.99  1971 1.00
## y_rep[383]   158.79    0.14 5.88   151.31   158.84   166.43  1875 1.00
## y_rep[384]    86.74    0.13 5.69    79.47    86.70    94.21  2013 1.00
## y_rep[385]    97.03    0.14 5.89    89.49    97.08   104.58  1777 1.00
## y_rep[386]   157.04    0.14 5.83   149.59   157.02   164.58  1731 1.00
## y_rep[387]   157.37    0.14 5.76   150.08   157.59   164.28  1643 1.00
## y_rep[388]   158.91    0.13 5.90   151.30   158.81   166.53  2158 1.00
## y_rep[389]   158.06    0.13 5.72   150.83   158.06   165.38  1856 1.00
## y_rep[390]   114.87    0.13 5.68   107.67   114.79   122.32  2034 1.00
## y_rep[391]   109.33    0.13 5.80   101.83   109.37   116.68  1913 1.00
## y_rep[392]    87.03    0.14 5.87    79.56    86.99    94.54  1783 1.00
## y_rep[393]   156.08    0.13 5.87   148.63   155.91   163.57  2086 1.00
## y_rep[394]   131.69    0.12 5.72   124.47   131.74   138.98  2217 1.00
## y_rep[395]   112.01    0.13 5.72   104.86   111.90   119.34  1827 1.00
## y_rep[396]    96.62    0.13 5.71    89.40    96.78   103.65  1836 1.00
## y_rep[397]   160.47    0.13 5.63   153.43   160.26   167.69  1935 1.00
## y_rep[398]   152.55    0.12 5.77   145.49   152.47   159.85  2158 1.00
## y_rep[399]   130.17    0.13 5.80   122.70   130.23   137.53  1987 1.00
## y_rep[400]   159.67    0.13 5.72   152.24   159.83   166.84  2002 1.00
## y_rep[401]    81.35    0.14 5.92    74.12    81.27    88.95  1811 1.00
## y_rep[402]    75.24    0.13 5.80    67.68    75.26    82.54  1881 1.00
## y_rep[403]   157.79    0.13 5.58   150.74   157.80   165.01  1788 1.00
## y_rep[404]   112.78    0.13 5.70   105.49   112.77   119.96  1953 1.00
## y_rep[405]   149.92    0.13 5.77   142.69   149.98   157.20  1895 1.00
## y_rep[406]   159.22    0.14 5.53   152.28   159.14   166.38  1626 1.00
## y_rep[407]    74.64    0.13 5.77    67.33    74.70    81.82  1887 1.00
## y_rep[408]   133.52    0.13 5.79   125.91   133.66   140.70  2076 1.00
## y_rep[409]    76.17    0.13 5.66    69.00    76.12    83.41  2002 1.00
## y_rep[410]   156.15    0.13 5.81   148.90   156.28   163.43  2115 1.00
## y_rep[411]   160.05    0.13 5.94   152.43   159.98   167.62  1980 1.00
## y_rep[412]   152.99    0.13 5.96   145.31   152.90   160.67  1961 1.00
## y_rep[413]   149.44    0.12 5.71   142.17   149.31   156.76  2168 1.00
## y_rep[414]   159.86    0.13 5.63   152.68   159.84   166.97  2001 1.00
## y_rep[415]   156.96    0.13 5.70   149.61   157.07   164.05  2027 1.00
## y_rep[416]    77.02    0.13 5.66    69.71    77.06    84.16  1974 1.00
## y_rep[417]   155.29    0.13 5.89   147.73   155.32   162.61  2022 1.00
## y_rep[418]   158.02    0.13 5.61   150.87   158.20   165.14  1965 1.00
## y_rep[419]   151.88    0.13 5.85   144.50   151.75   159.32  2169 1.00
## y_rep[420]   145.42    0.14 5.86   137.79   145.39   152.77  1850 1.00
## y_rep[421]   121.78    0.13 5.70   114.54   121.94   128.95  2078 1.00
## y_rep[422]   151.56    0.12 5.66   144.11   151.57   158.66  2320 1.00
## y_rep[423]   149.16    0.13 5.87   141.73   149.20   156.74  1995 1.00
## y_rep[424]   159.08    0.14 5.74   151.66   159.19   166.20  1614 1.00
## y_rep[425]   146.63    0.12 5.74   139.23   146.67   153.86  2193 1.00
## y_rep[426]    81.71    0.14 5.80    74.25    81.56    89.41  1823 1.00
## y_rep[427]   156.06    0.13 5.75   148.78   156.02   163.41  2012 1.00
## y_rep[428]   158.90    0.13 5.72   151.42   158.90   166.35  2018 1.00
## y_rep[429]   136.93    0.13 5.67   129.58   136.93   144.31  1949 1.00
## y_rep[430]   159.95    0.14 5.76   152.62   159.93   167.18  1768 1.00
## y_rep[431]   156.08    0.13 5.87   148.55   156.09   163.50  1951 1.00
## y_rep[432]   133.90    0.13 5.79   126.44   133.82   141.39  2004 1.00
## y_rep[433]   115.38    0.14 5.84   107.89   115.43   122.69  1789 1.00
## y_rep[434]   158.89    0.13 5.87   151.39   158.95   166.32  2037 1.00
## y_rep[435]   151.54    0.14 5.82   144.08   151.44   159.14  1712 1.00
## y_rep[436]   158.03    0.13 5.83   150.64   157.93   165.04  1873 1.00
## y_rep[437]   160.07    0.14 5.90   152.61   160.11   167.65  1876 1.00
## y_rep[438]   159.04    0.13 5.63   151.90   159.16   166.28  1954 1.00
## y_rep[439]   158.90    0.13 5.93   151.51   158.81   166.39  2196 1.00
## y_rep[440]    95.04    0.13 5.84    87.58    95.05   102.67  2135 1.00
## y_rep[441]    72.11    0.14 5.87    64.64    71.94    79.70  1878 1.00
## y_rep[442]   158.44    0.14 5.85   151.22   158.35   165.84  1845 1.00
## y_rep[443]   146.55    0.13 5.77   139.52   146.56   154.08  2008 1.00
## y_rep[444]   148.08    0.13 5.78   140.79   147.88   155.47  1998 1.00
## y_rep[445]   159.16    0.12 5.75   151.90   159.32   166.35  2147 1.00
## y_rep[446]   159.75    0.13 5.78   152.07   159.88   167.13  2069 1.00
## y_rep[447]   157.89    0.12 5.58   150.74   157.73   165.38  2026 1.00
## y_rep[448]   148.59    0.13 5.62   141.46   148.40   155.79  1784 1.00
## y_rep[449]   126.93    0.13 5.58   119.88   127.07   134.01  1824 1.00
## y_rep[450]   103.95    0.14 5.72    96.42   104.05   111.30  1710 1.00
## y_rep[451]   158.89    0.12 5.55   151.81   158.89   165.88  2062 1.00
## y_rep[452]   150.01    0.12 5.64   142.95   149.88   157.30  2133 1.00
## y_rep[453]    78.78    0.13 5.78    71.42    78.84    86.11  2020 1.00
## y_rep[454]   158.36    0.13 5.83   150.80   158.43   165.57  1984 1.00
## y_rep[455]   134.30    0.13 5.81   126.85   134.25   141.86  2002 1.00
## y_rep[456]    77.38    0.13 5.88    69.85    77.31    84.94  2030 1.00
## y_rep[457]    73.88    0.13 5.65    66.55    73.99    81.18  1843 1.00
## y_rep[458]   160.11    0.14 5.99   152.26   160.20   167.67  1953 1.00
## y_rep[459]   141.85    0.13 5.83   134.39   141.96   149.27  2148 1.00
## y_rep[460]   160.27    0.13 5.73   153.09   160.26   167.80  2003 1.00
## y_rep[461]   158.80    0.13 5.63   151.67   158.80   166.14  1972 1.00
## y_rep[462]   155.58    0.13 5.69   148.39   155.52   162.90  1941 1.00
## y_rep[463]    74.12    0.13 5.73    66.95    74.02    81.48  1909 1.00
## y_rep[464]   144.56    0.13 5.76   137.23   144.52   152.07  1977 1.00
## y_rep[465]   159.49    0.13 5.86   151.90   159.61   167.21  2007 1.00
## y_rep[466]   153.37    0.13 5.84   146.08   153.27   160.80  2017 1.00
## y_rep[467]    65.23    0.13 5.95    57.34    65.21    72.99  2060 1.00
## y_rep[468]    70.44    0.13 5.89    62.98    70.47    77.84  2111 1.00
## y_rep[469]   156.18    0.13 5.73   148.86   156.23   163.47  1989 1.00
## y_rep[470]   143.49    0.14 5.87   136.09   143.24   151.13  1761 1.00
## y_rep[471]   105.81    0.13 5.93    98.36   105.64   113.69  2063 1.00
## y_rep[472]    77.90    0.14 5.79    70.51    77.83    85.38  1650 1.00
## y_rep[473]   158.38    0.13 5.79   150.86   158.37   165.73  1973 1.00
## y_rep[474]   157.74    0.12 5.69   150.72   157.75   164.94  2098 1.00
## y_rep[475]   115.01    0.13 5.81   107.54   114.99   122.52  2124 1.00
## y_rep[476]    93.33    0.13 5.73    86.16    93.43   100.56  1999 1.00
## y_rep[477]   158.52    0.14 5.89   150.89   158.60   166.08  1863 1.00
## y_rep[478]   153.04    0.13 5.71   146.10   152.94   160.23  2025 1.00
## y_rep[479]   159.62    0.13 5.74   152.50   159.71   166.97  1935 1.00
## y_rep[480]   157.02    0.13 5.87   149.44   156.98   164.31  1951 1.00
## y_rep[481]    89.55    0.13 5.99    81.82    89.52    97.27  2018 1.00
## y_rep[482]   154.63    0.13 5.76   147.24   154.70   162.15  1904 1.00
## y_rep[483]   153.70    0.12 5.71   146.17   153.88   160.73  2188 1.00
## y_rep[484]    81.35    0.13 5.88    73.89    81.48    88.83  2016 1.00
## y_rep[485]   157.16    0.13 5.74   149.58   157.32   164.40  1927 1.00
## y_rep[486]   159.97    0.14 5.91   152.60   159.97   167.36  1876 1.00
## y_rep[487]   117.65    0.13 5.89   109.99   117.70   125.07  2153 1.00
## y_rep[488]    89.53    0.13 5.72    82.22    89.71    96.97  1932 1.00
## y_rep[489]   159.28    0.13 5.76   152.05   159.12   166.74  1834 1.00
## y_rep[490]   133.21    0.13 5.83   125.85   133.13   140.98  2074 1.00
## y_rep[491]   160.00    0.13 5.84   152.51   159.85   167.64  2012 1.00
## y_rep[492]   156.04    0.12 5.73   148.70   155.97   163.33  2260 1.00
## y_rep[493]   108.27    0.13 5.73   100.97   108.37   115.16  1801 1.00
## y_rep[494]   146.58    0.13 5.93   138.76   146.56   154.11  1937 1.00
## y_rep[495]   144.79    0.13 5.92   136.95   144.90   152.40  2032 1.00
## y_rep[496]   155.38    0.13 5.88   147.87   155.46   162.99  2003 1.00
## y_rep[497]   142.18    0.13 5.94   134.57   142.15   149.78  2098 1.00
## y_rep[498]   158.15    0.13 5.68   150.93   158.21   165.47  1898 1.00
## y_rep[499]   154.40    0.13 5.77   147.16   154.27   161.89  1931 1.00
## y_rep[500]   150.09    0.13 5.92   142.46   150.05   157.57  1925 1.00
## y_rep[501]   155.68    0.13 5.80   148.38   155.78   163.20  1887 1.00
## y_rep[502]   160.33    0.14 5.83   152.87   160.19   167.89  1773 1.00
## y_rep[503]   155.26    0.12 5.83   147.89   155.07   162.65  2252 1.00
## y_rep[504]   118.28    0.13 5.75   110.77   118.40   125.50  1934 1.00
## y_rep[505]    88.68    0.13 5.72    81.36    88.56    96.06  1919 1.00
## y_rep[506]   151.97    0.13 5.77   144.48   151.82   159.55  1873 1.00
## y_rep[507]   155.41    0.14 5.70   147.96   155.49   162.85  1725 1.00
## y_rep[508]   101.72    0.13 5.87    93.98   101.92   109.16  1956 1.00
## y_rep[509]   119.62    0.13 5.77   112.32   119.41   127.29  2013 1.00
## y_rep[510]   160.14    0.13 5.86   152.58   160.23   167.59  1912 1.00
## y_rep[511]   150.52    0.14 5.63   143.39   150.49   157.60  1697 1.00
## y_rep[512]   112.51    0.13 5.80   104.86   112.39   120.11  1905 1.00
## y_rep[513]    87.94    0.13 5.91    80.69    87.95    95.37  1931 1.00
## y_rep[514]    95.60    0.14 5.88    88.07    95.67   103.01  1894 1.00
## y_rep[515]   155.19    0.13 5.90   147.59   155.24   162.76  1968 1.00
## y_rep[516]   147.09    0.14 5.96   139.45   147.09   154.62  1938 1.00
## y_rep[517]   156.35    0.13 5.64   149.05   156.44   163.51  1959 1.00
## y_rep[518]    62.78    0.13 5.75    55.50    62.79    70.17  1949 1.00
## y_rep[519]   149.95    0.13 5.74   142.51   149.88   157.46  1839 1.00
## y_rep[520]    66.38    0.13 5.78    58.83    66.32    73.81  2072 1.00
## y_rep[521]    98.21    0.14 5.73    91.04    98.17   105.47  1776 1.00
## y_rep[522]   122.67    0.13 5.78   115.26   122.70   130.15  1869 1.00
## y_rep[523]    86.34    0.14 5.67    78.95    86.23    93.55  1597 1.00
## y_rep[524]   156.22    0.13 5.67   149.21   156.22   163.42  1960 1.00
## y_rep[525]   156.31    0.13 5.92   148.67   156.54   164.00  1956 1.00
## y_rep[526]   126.74    0.13 5.85   119.59   126.58   133.95  1896 1.00
## y_rep[527]   121.06    0.13 5.69   113.72   121.21   128.37  1998 1.00
## y_rep[528]    98.27    0.13 5.76    90.88    98.20   105.89  1958 1.00
## y_rep[529]   158.58    0.13 5.81   151.37   158.50   166.23  1959 1.00
## y_rep[530]   151.49    0.13 5.88   144.08   151.42   159.02  2117 1.00
## y_rep[531]   126.36    0.13 5.81   118.81   126.38   134.08  2118 1.00
## y_rep[532]    86.35    0.13 5.72    79.03    86.33    93.74  2086 1.00
## y_rep[533]   153.43    0.13 5.78   146.01   153.37   160.58  2019 1.00
## y_rep[534]   153.00    0.13 5.69   145.66   153.09   160.29  1909 1.00
## y_rep[535]   158.08    0.13 5.82   150.85   157.88   165.79  2125 1.00
## y_rep[536]   107.68    0.13 5.62   100.36   107.66   115.01  1827 1.00
## y_rep[537]    74.14    0.14 5.94    66.59    74.15    81.52  1911 1.00
## y_rep[538]   144.84    0.15 5.75   137.67   144.70   152.34  1543 1.00
## y_rep[539]    77.10    0.12 5.50    70.10    76.95    84.45  2129 1.00
## y_rep[540]   139.46    0.13 5.86   131.94   139.47   146.90  2057 1.00
## y_rep[541]   160.00    0.14 5.82   152.86   159.99   167.30  1740 1.00
## y_rep[542]   160.28    0.13 5.72   153.00   160.41   167.60  1830 1.00
## y_rep[543]    76.92    0.15 5.84    69.47    76.80    84.53  1470 1.00
## y_rep[544]   160.07    0.13 5.73   152.56   160.13   167.42  1995 1.00
## lp__       -1225.53    0.05 1.38 -1227.32 -1225.20 -1224.11   815 1.01
## 
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:57:43 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).
summary(lm(height~weight.z+I(weight.z^2),data=howell1))
## 
## Call:
## lm(formula = height ~ weight.z + I(weight.z^2), data = howell1)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -19.9689  -3.9794   0.2364   3.9262  19.5182 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   146.6604     0.3748  391.30   <2e-16 ***
## weight.z       21.4149     0.2908   73.64   <2e-16 ***
## I(weight.z^2)  -8.4123     0.2822  -29.80   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.766 on 541 degrees of freedom
## Multiple R-squared:  0.9565, Adjusted R-squared:  0.9564 
## F-statistic:  5952 on 2 and 541 DF,  p-value: < 2.2e-16

Figure 4.9 of Statistical rethinking

6 One continious variable + one categorical variable

6.1 Howell 2: subset with only individuals over 18 years old

ggplot(howell2, aes(y = height, x = weight, group = male)) + geom_point() + geom_smooth(method = "lm")

Xmat <- model.matrix(~weight.z + male, howell2)
m8.data <- with(howell2, list(y = height, X = Xmat, nX = ncol(Xmat), n = nrow(howell2)))
data {
  int<lower=1> n;
  int<lower=1> nX;
  vector [n] y;
  matrix [n,nX] X;
  }
parameters {
  vector[nX] beta;
  real sigma;
  }
  transformed parameters {
  vector[n] mu;

  mu = X*beta;
  }
model {
  y~normal(mu,sigma);
  
  beta ~ normal(0,100);
  sigma~cauchy(0,5);
  }
generated quantities {
  vector[n] log_lik;
  
  for (i in 1:n) {
  log_lik[i] = normal_lpdf(y[i] | mu[i], sigma); 
  }
  }
fit.m8 <- sampling(m8, data = m8.data, iter = 1000, chains = 4, cores = 4)
print(fit.m8, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: ff983e7d557e1f552144a41483b33c4a.
## 4 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=2000.
## 
##                 mean se_mean   sd     10%     50%     90% n_eff Rhat
## beta[1]       151.56    0.01 0.34  151.13  151.56  152.00  1318    1
## beta[2]         4.14    0.01 0.27    3.79    4.13    4.49  1313    1
## beta[3]         6.50    0.02 0.54    5.79    6.51    7.20  1188    1
## sigma           4.28    0.00 0.16    4.07    4.28    4.49  1917    1
## mu[1]         159.87    0.01 0.34  159.43  159.87  160.30  2005    1
## mu[2]         146.11    0.01 0.39  145.62  146.10  146.62  1832    1
## mu[3]         143.15    0.01 0.52  142.48  143.14  143.83  1729    1
## mu[4]         163.21    0.01 0.39  162.70  163.22  163.72  1971    1
## mu[5]         149.18    0.01 0.31  148.78  149.18  149.58  1612    1
## mu[6]         169.59    0.02 0.70  168.69  169.60  170.49  1482    1
## mu[7]         147.23    0.01 0.35  146.79  147.23  147.70  1812    1
## mu[8]         164.78    0.01 0.45  164.18  164.79  165.35  1717    1
## mu[9]         145.07    0.01 0.43  144.54  145.07  145.64  1808    1
## mu[10]        164.14    0.01 0.43  163.58  164.15  164.69  1809    1
## mu[11]        154.70    0.01 0.46  154.10  154.70  155.29  1190    1
## mu[12]        155.64    0.01 0.45  155.05  155.64  156.21  1400    1
## mu[13]        145.82    0.01 0.40  145.32  145.81  146.34  1828    1
## mu[14]        153.29    0.01 0.40  152.78  153.29  153.80  1220    1
## mu[15]        144.42    0.01 0.46  143.84  144.42  145.02  1783    1
## mu[16]        160.34    0.01 0.34  159.91  160.35  160.77  2082    1
## mu[17]        156.35    0.01 0.42  155.80  156.35  156.88  1455    1
## mu[18]        147.31    0.01 0.35  146.86  147.31  147.77  1808    1
## mu[19]        160.62    0.01 0.34  160.18  160.62  161.04  2119    1
## mu[20]        150.10    0.01 0.32  149.70  150.10  150.50  1481    1
## mu[21]        151.97    0.02 0.64  151.14  151.97  152.80  1279    1
## mu[22]        147.02    0.01 0.35  146.57  147.01  147.49  1821    1
## mu[23]        145.45    0.01 0.41  144.94  145.45  145.99  1819    1
## mu[24]        155.06    0.01 0.48  154.44  155.06  155.67  1366    1
## mu[25]        164.54    0.01 0.44  163.95  164.55  165.11  1749    1
## mu[26]        146.76    0.01 0.36  146.30  146.76  147.25  1828    1
## mu[27]        147.40    0.01 0.34  146.96  147.40  147.85  1803    1
## mu[28]        152.59    0.01 0.37  152.13  152.59  153.07  1249    1
## mu[29]        147.62    0.01 0.34  147.18  147.62  148.06  1789    1
## mu[30]        145.53    0.01 0.41  145.02  145.52  146.06  1821    1
## mu[31]        153.10    0.01 0.39  152.60  153.10  153.60  1227    1
## mu[32]        159.91    0.01 0.34  159.47  159.91  160.33  2011    1
## mu[33]        160.89    0.01 0.34  160.45  160.89  161.32  2150    1
## mu[34]        160.87    0.01 0.34  160.43  160.87  161.30  2148    1
## mu[35]        165.47    0.01 0.48  164.84  165.48  166.09  1647    1
## mu[36]        147.80    0.01 0.33  147.37  147.80  148.24  1775    1
## mu[37]        154.70    0.01 0.46  154.10  154.70  155.29  1190    1
## mu[38]        155.66    0.01 0.45  155.07  155.66  156.23  1401    1
## mu[39]        147.43    0.01 0.34  146.99  147.43  147.89  1801    1
## mu[40]        156.39    0.01 0.42  155.85  156.40  156.93  1460    1
## mu[41]        157.84    0.01 0.37  157.35  157.84  158.31  1640    1
## mu[42]        166.77    0.01 0.55  166.06  166.79  167.47  1577    1
## mu[43]        152.67    0.01 0.37  152.20  152.67  153.15  1246    1
## mu[44]        151.07    0.01 0.33  150.65  151.07  151.49  1365    1
## mu[45]        155.34    0.01 0.49  154.72  155.34  155.97  1185    1
## mu[46]        157.54    0.02 0.61  156.75  157.53  158.32  1174    1
## mu[47]        158.63    0.01 0.35  158.17  158.63  159.08  1781    1
## mu[48]        153.50    0.01 0.41  152.98  153.50  154.02  1214    1
## mu[49]        162.69    0.01 0.38  162.19  162.70  163.16  2072    1
## mu[50]        153.25    0.01 0.40  152.74  153.25  153.75  1222    1
## mu[51]        151.98    0.01 0.35  151.54  151.98  152.43  1286    1
## mu[52]        149.90    0.01 0.31  149.50  149.90  150.30  1508    1
## mu[53]        152.60    0.02 0.61  151.82  152.61  153.40  1289    1
## mu[54]        165.07    0.01 0.47  164.46  165.08  165.66  1679    1
## mu[55]        146.56    0.01 0.37  146.10  146.55  147.05  1831    1
## mu[56]        160.18    0.01 0.34  159.74  160.18  160.60  2057    1
## mu[57]        146.65    0.01 0.37  146.19  146.65  147.14  1830    1
## mu[58]        159.45    0.01 0.34  159.01  159.45  159.89  1930    1
## mu[59]        153.03    0.01 0.39  152.54  153.03  153.52  1230    1
## mu[60]        166.10    0.01 0.51  165.43  166.12  166.76  1610    1
## mu[61]        147.03    0.01 0.35  146.59  147.03  147.51  1820    1
## mu[62]        152.98    0.01 0.39  152.49  152.97  153.47  1232    1
## mu[63]        156.91    0.01 0.40  156.38  156.91  157.42  1512    1
## mu[64]        161.43    0.01 0.35  160.99  161.44  161.87  2189    1
## mu[65]        147.96    0.01 0.33  147.54  147.96  148.39  1760    1
## mu[66]        161.71    0.01 0.35  161.24  161.72  162.15  2197    1
## mu[67]        148.00    0.01 0.33  147.58  148.00  148.43  1757    1
## mu[68]        145.98    0.01 0.39  145.49  145.97  146.50  1830    1
## mu[69]        158.87    0.01 0.35  158.42  158.87  159.31  1823    1
## mu[70]        149.81    0.01 0.31  149.41  149.81  150.21  1521    1
## mu[71]        159.74    0.01 0.34  159.31  159.75  160.17  1983    1
## mu[72]        164.92    0.01 0.46  164.32  164.94  165.51  1698    1
## mu[73]        160.34    0.01 0.34  159.91  160.35  160.77  2082    1
## mu[74]        150.16    0.01 0.32  149.75  150.16  150.56  1474    1
## mu[75]        145.20    0.01 0.42  144.67  145.20  145.75  1812    1
## mu[76]        159.10    0.02 0.70  158.21  159.08  159.99  1177    1
## mu[77]        155.57    0.01 0.51  154.93  155.56  156.21  1183    1
## mu[78]        159.49    0.01 0.34  159.05  159.49  159.92  1936    1
## mu[79]        148.96    0.01 0.32  148.56  148.96  149.37  1643    1
## mu[80]        143.69    0.01 0.50  143.06  143.69  144.34  1752    1
## mu[81]        165.79    0.01 0.50  165.15  165.81  166.43  1627    1
## mu[82]        156.76    0.01 0.41  156.23  156.77  157.28  1496    1
## mu[83]        148.32    0.01 0.32  147.91  148.32  148.74  1723    1
## mu[84]        158.67    0.01 0.35  158.21  158.67  159.12  1787    1
## mu[85]        149.07    0.01 0.31  148.67  149.07  149.48  1627    1
## mu[86]        159.72    0.01 0.34  159.29  159.73  160.16  1979    1
## mu[87]        146.76    0.01 0.36  146.30  146.76  147.25  1828    1
## mu[88]        158.07    0.01 0.37  157.59  158.07  158.53  1680    1
## mu[89]        149.81    0.01 0.31  149.41  149.81  150.21  1521    1
## mu[90]        164.38    0.01 0.44  163.80  164.38  164.94  1773    1
## mu[91]        163.52    0.01 0.40  162.99  163.53  164.04  1914    1
## mu[92]        162.67    0.01 0.37  162.18  162.68  163.14  2075    1
## mu[93]        149.90    0.01 0.31  149.50  149.90  150.30  1508    1
## mu[94]        154.50    0.01 0.45  153.92  154.50  155.07  1193    1
## mu[95]        155.96    0.01 0.44  155.40  155.97  156.53  1424    1
## mu[96]        165.12    0.01 0.47  164.51  165.13  165.72  1673    1
## mu[97]        149.74    0.01 0.31  149.34  149.74  150.14  1532    1
## mu[98]        163.25    0.01 0.39  162.73  163.26  163.76  1964    1
## mu[99]        161.41    0.01 0.35  160.97  161.42  161.86  2188    1
## mu[100]       149.98    0.01 0.31  149.57  149.97  150.37  1498    1
## mu[101]       160.69    0.01 0.34  160.25  160.69  161.12  2128    1
## mu[102]       147.40    0.01 0.34  146.96  147.40  147.85  1803    1
## mu[103]       154.65    0.01 0.46  154.05  154.64  155.23  1191    1
## mu[104]       167.52    0.01 0.59  166.76  167.53  168.27  1545    1
## mu[105]       159.94    0.01 0.34  159.51  159.95  160.37  2018    1
## mu[106]       147.91    0.01 0.33  147.48  147.91  148.34  1765    1
## mu[107]       146.36    0.01 0.38  145.89  146.35  146.86  1832    1
## mu[108]       149.72    0.01 0.31  149.32  149.72  150.12  1534    1
## mu[109]       157.78    0.01 0.37  157.29  157.78  158.26  1631    1
## mu[110]       153.39    0.01 0.40  152.88  153.39  153.90  1217    1
## mu[111]       164.59    0.01 0.44  164.01  164.60  165.16  1741    1
## mu[112]       143.87    0.01 0.49  143.25  143.87  144.50  1760    1
## mu[113]       148.92    0.01 0.32  148.52  148.93  149.33  1647    1
## mu[114]       162.03    0.01 0.36  161.56  162.04  162.49  2194    1
## mu[115]       160.65    0.01 0.34  160.21  160.66  161.08  2124    1
## mu[116]       156.97    0.02 0.58  156.22  156.96  157.71  1175    1
## mu[117]       163.89    0.01 0.42  163.33  163.89  164.42  1851    1
## mu[118]       164.70    0.01 0.45  164.11  164.71  165.28  1726    1
## mu[119]       149.80    0.01 0.31  149.39  149.79  150.19  1524    1
## mu[120]       148.96    0.01 0.32  148.56  148.96  149.37  1643    1
## mu[121]       161.07    0.01 0.34  160.63  161.08  161.50  2167    1
## mu[122]       151.14    0.01 0.33  150.72  151.14  151.56  1357    1
## mu[123]       158.67    0.01 0.35  158.21  158.67  159.12  1787    1
## mu[124]       149.61    0.01 0.31  149.21  149.61  150.01  1550    1
## mu[125]       155.72    0.01 0.51  155.07  155.71  156.37  1181    1
## mu[126]       151.00    0.01 0.33  150.58  150.99  151.41  1372    1
## mu[127]       143.36    0.01 0.51  142.71  143.36  144.03  1738    1
## mu[128]       165.59    0.01 0.49  164.96  165.61  166.22  1639    1
## mu[129]       156.48    0.02 0.55  155.78  156.47  157.18  1176    1
## mu[130]       146.11    0.01 0.39  145.62  146.10  146.62  1832    1
## mu[131]       160.52    0.01 0.34  160.09  160.53  160.95  2107    1
## mu[132]       165.72    0.01 0.50  165.08  165.74  166.36  1632    1
## mu[133]       149.71    0.01 0.31  149.30  149.70  150.10  1537    1
## mu[134]       161.38    0.01 0.35  160.93  161.39  161.82  2187    1
## mu[135]       152.56    0.01 0.37  152.10  152.56  153.03  1251    1
## mu[136]       168.83    0.02 0.66  167.97  168.84  169.67  1502    1
## mu[137]       160.62    0.01 0.34  160.18  160.62  161.04  2119    1
## mu[138]       149.42    0.02 0.79  148.40  149.43  150.46  1260    1
## mu[139]       145.35    0.01 0.42  144.82  145.34  145.89  1816    1
## mu[140]       158.47    0.01 0.36  158.01  158.47  158.92  1753    1
## mu[141]       150.85    0.01 0.32  150.44  150.85  151.27  1388    1
## mu[142]       151.92    0.01 0.35  151.48  151.92  152.37  1290    1
## mu[143]       147.72    0.01 0.33  147.29  147.73  148.17  1781    1
## mu[144]       149.11    0.01 0.31  148.70  149.11  149.51  1622    1
## mu[145]       151.72    0.01 0.34  151.29  151.72  152.16  1305    1
## mu[146]       163.59    0.01 0.41  163.05  163.60  164.12  1901    1
## mu[147]       156.25    0.02 0.54  155.55  156.24  156.93  1177    1
## mu[148]       147.78    0.01 0.33  147.35  147.78  148.22  1776    1
## mu[149]       158.51    0.01 0.36  158.04  158.51  158.96  1759    1
## mu[150]       148.62    0.01 0.32  148.20  148.62  149.03  1688    1
## mu[151]       162.21    0.01 0.36  161.74  162.23  162.68  2161    1
## mu[152]       147.51    0.01 0.34  147.07  147.51  147.96  1796    1
## mu[153]       154.37    0.01 0.51  153.71  154.37  155.03  1336    1
## mu[154]       151.14    0.01 0.33  150.72  151.14  151.56  1357    1
## mu[155]       154.55    0.01 0.50  153.91  154.55  155.20  1343    1
## mu[156]       142.64    0.01 0.55  141.93  142.63  143.36  1708    1
## mu[157]       152.70    0.01 0.38  152.23  152.70  153.19  1244    1
## mu[158]       155.26    0.01 0.47  154.65  155.26  155.86  1377    1
## mu[159]       161.38    0.01 0.35  160.93  161.39  161.82  2187    1
## mu[160]       149.18    0.01 0.31  148.78  149.18  149.58  1612    1
## mu[161]       164.21    0.01 0.43  163.64  164.22  164.77  1798    1
## mu[162]       151.56    0.01 0.34  151.13  151.56  152.00  1318    1
## mu[163]       152.17    0.02 0.63  151.36  152.17  152.99  1282    1
## mu[164]       158.20    0.01 0.36  157.73  158.20  158.66  1703    1
## mu[165]       160.05    0.01 0.34  159.62  160.05  160.47  2036    1
## mu[166]       151.99    0.01 0.35  151.55  151.99  152.44  1285    1
## mu[167]       160.25    0.01 0.34  159.82  160.26  160.68  2068    1
## mu[168]       149.12    0.01 0.31  148.72  149.12  149.53  1620    1
## mu[169]       147.34    0.01 0.34  146.90  147.34  147.80  1806    1
## mu[170]       149.72    0.01 0.31  149.32  149.72  150.12  1534    1
## mu[171]       147.23    0.01 0.35  146.79  147.23  147.70  1812    1
## mu[172]       160.20    0.01 0.34  159.76  160.20  160.62  2060    1
## mu[173]       147.67    0.01 0.34  147.24  147.67  148.12  1785    1
## mu[174]       154.87    0.01 0.49  154.25  154.88  155.51  1357    1
## mu[175]       154.90    0.01 0.47  154.30  154.90  155.50  1188    1
## mu[176]       164.01    0.01 0.42  163.45  164.02  164.56  1830    1
## mu[177]       151.95    0.01 0.35  151.51  151.95  152.40  1288    1
## mu[178]       148.00    0.01 0.33  147.58  148.00  148.43  1757    1
## mu[179]       156.91    0.01 0.40  156.38  156.91  157.42  1512    1
## mu[180]       143.15    0.01 0.52  142.48  143.14  143.83  1729    1
## mu[181]       158.34    0.01 0.36  157.88  158.34  158.80  1730    1
## mu[182]       147.11    0.01 0.35  146.66  147.11  147.58  1817    1
## mu[183]       145.85    0.01 0.40  145.36  145.84  146.38  1828    1
## mu[184]       148.92    0.01 0.32  148.52  148.93  149.33  1647    1
## mu[185]       159.92    0.01 0.34  159.49  159.93  160.35  2014    1
## mu[186]       159.80    0.01 0.34  159.36  159.80  160.23  1992    1
## mu[187]       152.45    0.01 0.37  151.99  152.45  152.92  1257    1
## mu[188]       155.60    0.01 0.46  155.02  155.61  156.18  1398    1
## mu[189]       146.18    0.01 0.38  145.70  146.17  146.69  1832    1
## mu[190]       160.07    0.01 0.34  159.63  160.07  160.49  2039    1
## mu[191]       146.25    0.01 0.38  145.78  146.24  146.76  1832    1
## mu[192]       165.47    0.01 0.48  164.84  165.48  166.09  1647    1
## mu[193]       147.53    0.01 0.34  147.09  147.52  147.98  1795    1
## mu[194]       159.65    0.01 0.34  159.22  159.65  160.09  1966    1
## mu[195]       145.93    0.01 0.39  145.43  145.92  146.45  1829    1
## mu[196]       149.18    0.01 0.31  148.78  149.18  149.58  1612    1
## mu[197]       151.41    0.01 0.34  150.98  151.41  151.85  1331    1
## mu[198]       161.54    0.01 0.35  161.09  161.55  161.99  2193    1
## mu[199]       164.65    0.01 0.45  164.06  164.66  165.22  1734    1
## mu[200]       147.03    0.01 0.35  146.59  147.03  147.51  1820    1
## mu[201]       166.68    0.01 0.54  165.97  166.69  167.38  1581    1
## mu[202]       151.19    0.01 0.33  150.77  151.19  151.62  1352    1
## mu[203]       157.68    0.01 0.38  157.18  157.68  158.16  1614    1
## mu[204]       151.29    0.01 0.33  150.86  151.28  151.71  1343    1
## mu[205]       157.82    0.01 0.37  157.33  157.82  158.29  1637    1
## mu[206]       148.67    0.01 0.32  148.26  148.67  149.08  1681    1
## mu[207]       150.98    0.01 0.33  150.56  150.97  151.40  1374    1
## mu[208]       144.78    0.01 0.44  144.23  144.78  145.36  1797    1
## mu[209]       143.73    0.01 0.49  143.10  143.73  144.37  1754    1
## mu[210]       145.76    0.01 0.40  145.26  145.75  146.29  1827    1
## mu[211]       160.98    0.01 0.34  160.54  160.98  161.41  2159    1
## mu[212]       163.30    0.01 0.39  162.78  163.31  163.81  1954    1
## mu[213]       152.98    0.02 0.58  152.23  152.99  153.75  1296    1
## mu[214]       148.69    0.01 0.32  148.27  148.69  149.10  1679    1
## mu[215]       161.65    0.01 0.35  161.19  161.66  162.10  2196    1
## mu[216]       157.40    0.01 0.39  156.89  157.40  157.89  1574    1
## mu[217]       161.19    0.01 0.34  160.75  161.20  161.63  2176    1
## mu[218]       144.26    0.01 0.47  143.66  144.26  144.86  1776    1
## mu[219]       157.33    0.01 0.39  156.81  157.33  157.82  1564    1
## mu[220]       153.57    0.01 0.41  153.04  153.57  154.09  1212    1
## mu[221]       150.12    0.01 0.32  149.72  150.12  150.52  1478    1
## mu[222]       144.45    0.01 0.46  143.88  144.46  145.05  1784    1
## mu[223]       154.67    0.01 0.50  154.04  154.68  155.32  1348    1
## mu[224]       145.76    0.01 0.40  145.26  145.75  146.29  1827    1
## mu[225]       161.85    0.01 0.35  161.38  161.86  162.30  2197    1
## mu[226]       152.05    0.01 0.35  151.61  152.05  152.51  1281    1
## mu[227]       154.27    0.01 0.44  153.70  154.26  154.82  1197    1
## mu[228]       166.92    0.01 0.56  166.19  166.93  167.63  1570    1
## mu[229]       150.58    0.01 0.32  150.16  150.57  150.99  1420    1
## mu[230]       162.52    0.01 0.37  162.03  162.54  163.00  2103    1
## mu[231]       161.71    0.01 0.35  161.24  161.72  162.15  2197    1
## mu[232]       144.67    0.01 0.45  144.11  144.67  145.26  1793    1
## mu[233]       147.96    0.01 0.33  147.54  147.96  148.39  1760    1
## mu[234]       167.39    0.01 0.58  166.64  167.40  168.13  1550    1
## mu[235]       162.65    0.01 0.37  162.16  162.66  163.12  2079    1
## mu[236]       148.34    0.01 0.32  147.93  148.34  148.76  1721    1
## mu[237]       157.38    0.01 0.39  156.87  157.38  157.88  1572    1
## mu[238]       157.72    0.02 0.62  156.92  157.71  158.52  1174    1
## mu[239]       151.61    0.01 0.34  151.18  151.61  152.05  1314    1
## mu[240]       149.21    0.01 0.31  148.81  149.21  149.62  1607    1
## mu[241]       149.58    0.01 0.31  149.17  149.58  149.97  1555    1
## mu[242]       156.47    0.01 0.42  155.93  156.48  157.00  1467    1
## mu[243]       156.87    0.01 0.40  156.35  156.88  157.38  1508    1
## mu[244]       148.89    0.01 0.32  148.48  148.89  149.30  1652    1
## mu[245]       156.69    0.01 0.41  156.16  156.69  157.21  1489    1
## mu[246]       152.34    0.01 0.36  151.88  152.34  152.81  1263    1
## mu[247]       159.89    0.01 0.34  159.45  159.89  160.31  2008    1
## mu[248]       149.98    0.01 0.31  149.57  149.97  150.37  1498    1
## mu[249]       160.31    0.01 0.34  159.87  160.31  160.73  2076    1
## mu[250]       158.62    0.01 0.35  158.15  158.62  159.06  1778    1
## mu[251]       163.12    0.01 0.39  162.61  163.13  163.62  1988    1
## mu[252]       159.71    0.01 0.34  159.27  159.71  160.14  1976    1
## mu[253]       150.63    0.01 0.32  150.22  150.63  151.04  1413    1
## mu[254]       150.56    0.01 0.32  150.15  150.56  150.97  1422    1
## mu[255]       156.83    0.02 0.57  156.09  156.82  157.55  1175    1
## mu[256]       151.36    0.01 0.34  150.93  151.36  151.79  1336    1
## mu[257]       159.47    0.01 0.34  159.03  159.47  159.90  1933    1
## mu[258]       143.02    0.01 0.53  142.34  143.01  143.72  1724    1
## mu[259]       165.39    0.01 0.48  164.77  165.40  166.01  1651    1
## mu[260]       164.98    0.01 0.46  164.38  164.99  165.57  1691    1
## mu[261]       159.65    0.01 0.34  159.22  159.65  160.09  1966    1
## mu[262]       151.67    0.01 0.34  151.23  151.67  152.10  1309    1
## mu[263]       151.90    0.01 0.35  151.47  151.90  152.35  1291    1
## mu[264]       154.05    0.01 0.43  153.49  154.05  154.59  1201    1
## mu[265]       159.07    0.01 0.35  158.62  159.07  159.51  1859    1
## mu[266]       150.85    0.01 0.32  150.44  150.85  151.27  1388    1
## mu[267]       165.14    0.01 0.47  164.53  165.15  165.74  1671    1
## mu[268]       148.42    0.01 0.32  148.00  148.42  148.83  1712    1
## mu[269]       155.28    0.01 0.49  154.66  155.28  155.90  1185    1
## mu[270]       158.89    0.01 0.35  158.44  158.89  159.33  1827    1
## mu[271]       160.85    0.01 0.34  160.41  160.86  161.28  2147    1
## mu[272]       151.00    0.01 0.33  150.58  150.99  151.41  1372    1
## mu[273]       161.92    0.01 0.36  161.45  161.93  162.37  2197    1
## mu[274]       155.35    0.01 0.47  154.75  155.35  155.94  1382    1
## mu[275]       146.80    0.01 0.36  146.34  146.79  147.28  1827    1
## mu[276]       162.29    0.01 0.36  161.81  162.30  162.75  2148    1
## mu[277]       151.45    0.01 0.34  151.02  151.45  151.89  1328    1
## mu[278]       150.54    0.01 0.32  150.13  150.54  150.95  1424    1
## mu[279]       159.16    0.01 0.35  158.72  159.16  159.60  1876    1
## mu[280]       148.07    0.01 0.33  147.65  148.07  148.50  1750    1
## mu[281]       145.02    0.01 0.43  144.48  145.01  145.59  1806    1
## mu[282]       154.40    0.01 0.51  153.75  154.41  155.06  1337    1
## mu[283]       146.73    0.01 0.36  146.27  146.72  147.21  1829    1
## mu[284]       160.42    0.01 0.34  159.98  160.42  160.84  2092    1
## mu[285]       145.54    0.01 0.41  145.03  145.54  146.08  1822    1
## mu[286]       157.24    0.01 0.39  156.72  157.24  157.73  1552    1
## mu[287]       160.11    0.01 0.34  159.67  160.11  160.53  2045    1
## mu[288]       161.89    0.01 0.36  161.42  161.90  162.34  2197    1
## mu[289]       150.92    0.01 0.33  150.51  150.92  151.34  1380    1
## mu[290]       160.11    0.01 0.34  159.67  160.11  160.53  2045    1
## mu[291]       159.14    0.01 0.35  158.70  159.15  159.58  1873    1
## mu[292]       165.37    0.01 0.48  164.75  165.39  165.99  1653    1
## mu[293]       160.36    0.01 0.34  159.93  160.37  160.78  2085    1
## mu[294]       153.63    0.01 0.41  153.10  153.63  154.15  1210    1
## mu[295]       159.65    0.01 0.34  159.22  159.65  160.09  1966    1
## mu[296]       145.51    0.01 0.41  145.00  145.50  146.04  1821    1
## mu[297]       146.18    0.01 0.38  145.70  146.17  146.69  1832    1
## mu[298]       154.10    0.01 0.43  153.54  154.10  154.65  1200    1
## mu[299]       162.07    0.01 0.36  161.59  162.08  162.53  2187    1
## mu[300]       159.03    0.01 0.35  158.59  159.03  159.47  1853    1
## mu[301]       146.42    0.01 0.37  145.95  146.41  146.91  1832    1
## mu[302]       160.40    0.01 0.34  159.96  160.40  160.82  2090    1
## mu[303]       147.11    0.01 0.35  146.66  147.11  147.58  1817    1
## mu[304]       159.53    0.01 0.34  159.09  159.52  159.96  1943    1
## mu[305]       164.72    0.01 0.45  164.13  164.73  165.30  1724    1
## mu[306]       163.96    0.01 0.42  163.40  163.97  164.50  1838    1
## mu[307]       160.29    0.01 0.34  159.86  160.29  160.72  2074    1
## mu[308]       150.32    0.01 0.32  149.91  150.32  150.72  1452    1
## mu[309]       144.64    0.01 0.45  144.07  144.63  145.22  1792    1
## mu[310]       154.74    0.01 0.46  154.14  154.73  155.33  1190    1
## mu[311]       151.14    0.01 0.33  150.72  151.14  151.56  1357    1
## mu[312]       144.16    0.01 0.47  143.56  144.16  144.78  1772    1
## mu[313]       159.53    0.01 0.34  159.09  159.52  159.96  1943    1
## mu[314]       152.28    0.01 0.36  151.83  152.29  152.75  1266    1
## mu[315]       159.60    0.01 0.34  159.16  159.60  160.03  1956    1
## mu[316]       148.91    0.01 0.32  148.50  148.91  149.32  1650    1
## mu[317]       161.49    0.01 0.35  161.04  161.50  161.93  2191    1
## mu[318]       151.65    0.01 0.34  151.21  151.65  152.08  1311    1
## mu[319]       156.29    0.01 0.43  155.74  156.30  156.84  1450    1
## mu[320]       149.41    0.01 0.31  149.01  149.41  149.81  1578    1
## mu[321]       158.20    0.01 0.36  157.73  158.20  158.66  1703    1
## mu[322]       155.57    0.01 0.51  154.93  155.56  156.21  1183    1
## mu[323]       160.80    0.01 0.34  160.35  160.80  161.22  2141    1
## mu[324]       162.03    0.01 0.36  161.56  162.04  162.49  2194    1
## mu[325]       150.83    0.01 0.32  150.42  150.83  151.25  1390    1
## mu[326]       145.49    0.01 0.41  144.98  145.49  146.03  1820    1
## mu[327]       150.21    0.01 0.32  149.80  150.21  150.61  1467    1
## mu[328]       159.20    0.01 0.34  158.75  159.20  159.63  1883    1
## mu[329]       156.06    0.01 0.44  155.49  156.06  156.61  1431    1
## mu[330]       153.70    0.02 0.55  153.00  153.71  154.42  1314    1
## mu[331]       156.97    0.01 0.40  156.45  156.97  157.48  1520    1
## mu[332]       163.59    0.01 0.41  163.05  163.60  164.12  1901    1
## mu[333]       150.27    0.01 0.32  149.86  150.27  150.67  1459    1
## mu[334]       154.71    0.01 0.50  154.08  154.72  155.36  1350    1
## mu[335]       150.50    0.01 0.32  150.09  150.50  150.91  1429    1
## mu[336]       162.98    0.01 0.38  162.47  162.98  163.47  2016    1
## mu[337]       147.36    0.01 0.34  146.92  147.36  147.82  1805    1
## mu[338]       150.16    0.01 0.32  149.75  150.16  150.56  1474    1
## mu[339]       145.73    0.01 0.40  145.22  145.72  146.25  1826    1
## mu[340]       151.07    0.01 0.33  150.65  151.07  151.49  1365    1
## mu[341]       147.13    0.01 0.35  146.68  147.12  147.59  1817    1
## mu[342]       157.49    0.01 0.38  156.99  157.49  157.98  1587    1
## mu[343]       150.94    0.01 0.33  150.52  150.94  151.36  1378    1
## mu[344]       159.91    0.01 0.34  159.47  159.91  160.33  2011    1
## mu[345]       147.98    0.01 0.33  147.56  147.98  148.41  1759    1
## mu[346]       155.53    0.01 0.46  154.94  155.54  156.11  1393    1
## mu[347]       148.89    0.01 0.32  148.48  148.89  149.30  1652    1
## mu[348]       152.87    0.01 0.38  152.39  152.87  153.35  1237    1
## mu[349]       144.67    0.01 0.45  144.11  144.67  145.26  1793    1
## mu[350]       162.65    0.01 0.37  162.16  162.66  163.12  2079    1
## mu[351]       157.37    0.02 0.60  156.60  157.37  158.15  1174    1
## mu[352]       162.89    0.01 0.38  162.38  162.89  163.37  2033    1
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## log_lik[5]     -2.76    0.00 0.07   -2.85   -2.76   -2.68  1565    1
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## log_lik[13]    -2.41    0.00 0.04   -2.46   -2.40   -2.35  1718    1
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## log_lik[33]    -4.66    0.00 0.22   -4.94   -4.65   -4.38  1949    1
## log_lik[34]    -2.48    0.00 0.05   -2.55   -2.48   -2.42  1919    1
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## log_lik[36]    -2.38    0.00 0.04   -2.43   -2.38   -2.33  1833    1
## log_lik[37]    -3.62    0.00 0.18   -3.85   -3.62   -3.40  1316    1
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## log_lik[40]    -4.95    0.01 0.28   -5.31   -4.94   -4.60  1480    1
## log_lik[41]    -2.38    0.00 0.04   -2.43   -2.38   -2.33  1950    1
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## log_lik[73]    -5.03    0.00 0.24   -5.34   -5.02   -4.72  2339    1
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## log_lik[335]   -2.91    0.00 0.08   -3.02   -2.91   -2.82  1449    1
## log_lik[336]   -2.69    0.00 0.07   -2.79   -2.69   -2.60  2039    1
## log_lik[337]   -2.40    0.00 0.04   -2.45   -2.40   -2.35  1928    1
## log_lik[338]   -2.45    0.00 0.04   -2.50   -2.44   -2.39  1802    1
## log_lik[339]   -2.60    0.00 0.07   -2.69   -2.60   -2.52  1961    1
## log_lik[340]   -2.56    0.00 0.05   -2.63   -2.56   -2.50  1551    1
## log_lik[341]   -2.38    0.00 0.04   -2.43   -2.38   -2.33  1831    1
## log_lik[342]   -2.55    0.00 0.06   -2.63   -2.55   -2.48  1834    1
## log_lik[343]   -3.14    0.00 0.10   -3.26   -3.13   -3.02  1374    1
## log_lik[344]   -2.39    0.00 0.04   -2.44   -2.39   -2.34  1840    1
## log_lik[345]   -2.48    0.00 0.05   -2.54   -2.48   -2.42  1639    1
## log_lik[346]   -2.39    0.00 0.04   -2.44   -2.39   -2.34  1570    1
## log_lik[347]   -2.71    0.00 0.06   -2.79   -2.71   -2.64  1742    1
## log_lik[348]   -4.95    0.01 0.26   -5.29   -4.95   -4.62  1343    1
## log_lik[349]   -2.47    0.00 0.05   -2.54   -2.46   -2.40  1666    1
## log_lik[350]   -2.38    0.00 0.04   -2.42   -2.38   -2.33  1879    1
## log_lik[351]   -2.42    0.00 0.05   -2.49   -2.41   -2.35  1113    1
## log_lik[352]   -2.85    0.00 0.09   -2.96   -2.84   -2.74  2055    1
## lp__         -689.37    0.05 1.42 -691.22 -689.07 -687.86   818    1
## 
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:58:53 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).
summary(lm(height~weight.z+male,data=howell2))
## 
## Call:
## lm(formula = height ~ weight.z + male, data = howell2)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -21.7859  -2.5506   0.4669   2.6278  14.1486 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 151.5501     0.3391  446.98   <2e-16 ***
## weight.z      4.1399     0.2678   15.46   <2e-16 ***
## male          6.5003     0.5359   12.13   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.272 on 349 degrees of freedom
## Multiple R-squared:  0.6973, Adjusted R-squared:  0.6955 
## F-statistic: 401.9 on 2 and 349 DF,  p-value: < 2.2e-16

6.2 Howell 1: data set with all individuals

howell1$male <- ifelse(howell1$male==1,2,1)
m8.1.data <- list(N = nrow(howell1), height = howell1$height, weight = howell1$weight.z, male = as.integer(howell1$male), n = length(unique(howell1$male)))
data {                                                     // observed variables 
  int<lower=1> N;                                          // number of observations
  vector[N] height;                                        // response variables
  vector[N] weight;                                        // predictor (previously centered)
  int<lower=0> n; 
  int<lower=1,upper=n> male[N]; 
}

parameters {                                               // unobserved variables
  real beta[2];                                            // parameters: intercept and slope
  vector[n] alpha; 
  real sigma;
}

transformed parameters {
vector[N] mu;                                              // create one variable with predictions for each obs
for (i in 1:N)                                             // loop for all cases 
mu[i] = alpha[male[i]] + beta[1]*weight[i] + beta[2]*(weight[i] .* weight[i]);  
}


model {
  height ~ normal(mu,sigma);        // likelihood
  alpha ~ normal(178, 100);                              // prior of the intercept
  beta[1] ~ normal(0, 100);                                  // prior of the slope
  beta[2] ~ normal(0, 100);
  sigma ~ uniform(0,50);
}
fit.m8.1 <- sampling(m8.1, data = m8.1.data, iter = 1000, chains = 4, cores = 4)
## Warning: There were 275 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
## http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
## Warning: There were 1 chains where the estimated Bayesian Fraction of Missing Information was low. See
## http://mc-stan.org/misc/warnings.html#bfmi-low
## Warning: Examine the pairs() plot to diagnose sampling problems
print(fit.m8.1, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: 7f6f4063a593c9d5515aafcd1d091662.
## 4 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=2000.
## 
##              mean se_mean     sd      10%      50%      90% n_eff Rhat
## beta[1]     20.80    0.07   1.04    20.16    20.73    21.32   222 1.03
## beta[2]     -8.91    0.12   1.56    -9.56    -9.06    -8.31   179 1.04
## alpha[1]   144.86    0.27   3.24   143.62   145.26   145.89   142 1.04
## alpha[2]   149.38    0.18   2.32   148.37   149.56   150.49   166 1.04
## sigma        9.95    5.57   8.49     5.22     5.48    21.35     2 3.03
## mu[1]      160.50    0.05   0.93   159.77   160.53   161.23   366 1.00
## mu[2]      146.06    0.27   3.19   144.87   146.47   147.08   140 1.04
## mu[3]      138.99    0.28   3.35   137.77   139.39   140.05   145 1.04
## mu[4]      161.52    0.08   1.40   160.52   161.41   162.48   325 1.02
## mu[5]      151.54    0.28   2.74   150.43   151.89   152.47    99 1.04
## mu[6]      157.25    0.36   4.98   154.89   156.77   159.34   195 1.04
## mu[7]      148.29    0.26   3.06   147.16   148.67   149.29   137 1.04
## mu[8]      161.23    0.13   2.09   159.98   161.03   162.50   272 1.03
## mu[9]      143.79    0.27   3.28   142.55   144.20   144.83   143 1.04
## mu[10]     161.40    0.10   1.79   160.24   161.24   162.49   291 1.03
## mu[11]     156.66    0.14   1.53   155.59   156.84   157.58   121 1.04
## mu[12]     156.01    0.14   1.80   155.05   156.15   156.93   176 1.03
## mu[13]     145.45    0.27   3.22   144.24   145.85   146.47   141 1.04
## mu[14]     155.93    0.18   1.82   154.99   156.16   156.81   105 1.04
## mu[15]     142.24    0.28   3.32   141.02   142.65   143.29   144 1.04
## mu[16]     160.78    0.04   0.88   160.04   160.79   161.53   412 1.00
## mu[17]     157.01    0.12   1.66   156.11   157.13   157.90   183 1.03
## mu[18]     148.43    0.26   3.05   147.30   148.81   149.42   137 1.04
## mu[19]     116.26    0.13   1.69   115.29   116.36   117.32   173 1.03
## mu[20]      94.95    0.09   1.61    93.99    95.08    95.95   328 1.02
## mu[21]      83.41    0.05   1.40    82.34    83.46    84.61   755 1.00
## mu[22]     160.92    0.04   0.87   160.16   160.93   161.67   436 1.00
## mu[23]     152.83    0.29   2.56   151.78   153.15   153.73    78 1.04
## mu[24]     126.44    0.16   2.10   125.47   126.60   127.53   166 1.03
## mu[25]     101.30    0.12   1.91   100.20   101.50   102.25   267 1.03
## mu[26]     149.21    0.18   2.33   148.20   149.39   150.33   166 1.04
## mu[27]     147.88    0.26   3.09   146.80   148.26   148.88   138 1.04
## mu[28]     144.65    0.27   3.25   143.40   145.06   145.69   142 1.04
## mu[29]     134.85    0.18   2.36   133.88   135.02   135.99   165 1.04
## mu[30]     119.71    0.22   2.85   118.51   120.04   120.72   161 1.04
## mu[31]     111.34    0.11   1.50   110.34   111.39   112.36   180 1.02
## mu[32]     155.11    0.15   1.91   154.11   155.26   156.06   173 1.03
## mu[33]     161.30    0.12   1.97   160.08   161.12   162.47   281 1.03
## mu[34]     147.39    0.27   3.12   146.28   147.78   148.39   139 1.04
## mu[35]     148.60    0.26   3.04   147.47   148.97   149.59   136 1.04
## mu[36]     155.43    0.21   1.99   154.48   155.67   156.30    92 1.04
## mu[37]     148.99    0.26   3.01   147.87   149.36   149.97   135 1.04
## mu[38]     144.81    0.27   3.24   143.57   145.22   145.85   142 1.04
## mu[39]     155.80    0.18   1.86   154.88   156.03   156.68   103 1.04
## mu[40]     160.53    0.05   0.92   159.79   160.55   161.25   369 1.00
## mu[41]     161.05    0.04   0.87   160.26   161.05   161.80   452 1.00
## mu[42]     161.04    0.04   0.87   160.26   161.04   161.80   452 1.00
## mu[43]     160.94    0.16   2.44   159.58   160.71   162.31   240 1.03
## mu[44]     149.31    0.26   2.98   148.21   149.68   150.29   135 1.04
## mu[45]     156.66    0.14   1.53   155.59   156.84   157.58   121 1.04
## mu[46]     133.33    0.27   3.30   132.11   133.72   134.41   147 1.04
## mu[47]     118.40    0.14   1.78   117.41   118.52   119.49   171 1.03
## mu[48]     121.38    0.23   2.92   120.18   121.71   122.39   158 1.04
## mu[49]      92.76    0.08   1.53    91.78    92.87    93.80   367 1.02
## mu[50]     156.04    0.14   1.80   155.08   156.17   156.95   176 1.03
## mu[51]     148.66    0.26   3.03   147.53   149.04   149.65   136 1.04
## mu[52]     157.08    0.12   1.65   156.18   157.20   157.96   183 1.03
## mu[53]     158.79    0.09   1.33   157.98   158.87   159.61   220 1.02
## mu[54]     122.55    0.15   1.95   121.55   122.70   123.63   168 1.03
## mu[55]      99.57    0.11   1.82    98.53    99.75   100.53   280 1.03
## mu[56]      91.09    0.07   1.48    90.10    91.19    92.14   408 1.01
## mu[57]     160.14    0.22   3.17   158.49   159.83   161.66   216 1.04
## mu[58]     155.48    0.20   1.97   154.53   155.72   156.36    94 1.04
## mu[59]     153.98    0.29   2.35   152.97   154.27   154.86    65 1.04
## mu[60]     156.85    0.12   1.44   155.70   157.01   157.83   156 1.04
## mu[61]     156.91    0.12   1.56   155.46   156.96   158.17   157 1.04
## mu[62]     159.56    0.07   1.15   158.84   159.62   160.32   278 1.02
## mu[63]     156.06    0.17   1.77   155.15   156.28   156.95   106 1.04
## mu[64]     161.51    0.06   1.21   160.60   161.43   162.43   351 1.02
## mu[65]     155.90    0.18   1.83   154.96   156.13   156.77   104 1.04
## mu[66]     154.90    0.25   2.13   153.89   155.17   155.76    75 1.04
## mu[67]     152.57    0.29   2.60   151.55   152.89   153.47    82 1.04
## mu[68]     150.58    0.18   2.26   149.56   150.76   151.66   166 1.03
## mu[69]     161.12    0.14   2.23   159.82   160.91   162.43   256 1.03
## mu[70]     146.99    0.27   3.14   145.86   147.39   147.99   139 1.04
## mu[71]     128.72    0.17   2.18   127.74   128.89   129.82   166 1.03
## mu[72]     126.65    0.25   3.12   125.40   127.02   127.71   150 1.04
## mu[73]     160.69    0.04   0.89   159.95   160.71   161.43   396 1.00
## mu[74]     147.17    0.27   3.13   146.05   147.57   148.17   139 1.04
## mu[75]     134.87    0.27   3.33   133.66   135.26   135.95   147 1.04
## mu[76]     160.22    0.05   0.99   159.50   160.26   160.93   331 1.01
## mu[77]     155.75    0.19   1.88   154.81   155.98   156.63   102 1.04
## mu[78]      96.17    0.09   1.33    95.09    96.11    97.18   226 1.02
## mu[79]     160.59    0.19   2.79   159.10   160.33   162.04   226 1.04
## mu[80]     147.92    0.26   3.09   146.83   148.30   148.91   138 1.04
## mu[81]     152.12    0.28   2.67   151.06   152.45   153.03    89 1.04
## mu[82]     130.87    0.27   3.25   129.58   131.24   131.94   148 1.04
## mu[83]     155.71    0.19   1.89   154.77   155.95   156.59   102 1.04
## mu[84]     157.74    0.11   1.54   156.89   157.84   158.59   191 1.03
## mu[85]     161.26    0.04   0.92   160.47   161.23   162.03   446 1.00
## mu[86]     149.60    0.26   2.95   148.49   149.96   150.56   134 1.04
## mu[87]     110.19    0.11   1.46   109.17   110.24   111.20   183 1.02
## mu[88]     161.34    0.05   0.96   160.53   161.30   162.13   428 1.01
## mu[89]     149.66    0.26   2.95   148.55   150.02   150.62   134 1.04
## mu[90]     145.79    0.27   3.20   144.60   146.20   146.82   141 1.04
## mu[91]     159.77    0.06   1.10   159.04   159.82   160.51   292 1.01
## mu[92]     152.44    0.29   2.62   151.43   152.77   153.35    83 1.04
## mu[93]     160.42    0.05   0.95   159.69   160.46   161.14   354 1.01
## mu[94]     161.17    0.13   2.16   159.90   160.96   162.48   264 1.03
## mu[95]     109.55    0.16   2.34   108.38   109.81   110.50   204 1.03
## mu[96]     160.78    0.04   0.88   160.04   160.79   161.53   412 1.00
## mu[97]     152.90    0.29   2.55   151.85   153.22   153.80    77 1.04
## mu[98]     144.08    0.27   3.27   142.82   144.49   145.11   143 1.04
## mu[99]     111.42    0.11   1.50   110.42   111.47   112.44   180 1.03
## mu[100]    156.36    0.16   2.08   154.59   156.33   157.82   160 1.04
## mu[101]    135.09    0.18   2.36   134.12   135.26   136.23   164 1.04
## mu[102]    156.90    0.11   1.42   155.76   157.05   157.89   157 1.04
## mu[103]    160.25    0.05   0.99   159.52   160.29   160.96   334 1.01
## mu[104]    151.22    0.27   2.79   150.10   151.57   152.15   105 1.04
## mu[105]    140.42    0.28   3.34   139.19   140.83   141.48   145 1.04
## mu[106]    160.77    0.17   2.62   159.34   160.53   162.17   232 1.04
## mu[107]    157.56    0.11   1.57   156.70   157.67   158.42   189 1.03
## mu[108]    150.21    0.26   2.90   149.06   150.57   151.16   124 1.04
## mu[109]    140.00    0.28   3.35   138.77   140.40   141.06   145 1.04
## mu[110]    109.20    0.11   1.43   108.15   109.24   110.21   185 1.02
## mu[111]    107.35    0.10   1.38   106.30   107.39   108.37   190 1.02
## mu[112]    159.59    0.07   1.15   158.87   159.65   160.35   280 1.02
## mu[113]    151.38    0.27   2.77   150.28   151.73   152.31   102 1.04
## mu[114]    107.95    0.15   2.26   106.82   108.19   108.90   219 1.03
## mu[115]    103.04    0.09   1.29   102.01   103.05   104.06   206 1.02
## mu[116]     91.34    0.09   1.49    90.22    91.24    92.40   249 1.02
## mu[117]    160.41    0.05   0.95   159.68   160.44   161.12   353 1.01
## mu[118]    147.39    0.27   3.12   146.28   147.78   148.39   139 1.04
## mu[119]    159.03    0.08   1.28   158.25   159.11   159.83   234 1.02
## mu[120]    152.44    0.29   2.62   151.43   152.77   153.35    83 1.04
## mu[121]    161.35    0.11   1.90   160.16   161.17   162.50   285 1.03
## mu[122]    161.50    0.09   1.52   160.45   161.37   162.50   311 1.03
## mu[123]    161.51    0.06   1.21   160.60   161.43   162.42   352 1.02
## mu[124]     85.35    0.12   1.81    84.11    85.17    86.60   210 1.02
## mu[125]    152.57    0.29   2.60   151.55   152.89   153.47    82 1.04
## mu[126]    156.58    0.15   1.56   155.55   156.76   157.50   115 1.04
## mu[127]     96.73    0.09   1.32    95.66    96.67    97.74   226 1.02
## mu[128]    156.48    0.13   1.74   155.55   156.62   157.38   179 1.03
## mu[129]    161.09    0.14   2.26   159.79   160.88   162.42   253 1.03
## mu[130]    152.35    0.29   2.63   151.33   152.68   153.25    85 1.04
## mu[131]    111.12    0.17   2.42   109.94   111.39   112.07   193 1.03
## mu[132]    103.04    0.09   1.29   102.01   103.05   104.06   206 1.02
## mu[133]    161.52    0.08   1.41   160.51   161.41   162.48   323 1.02
## mu[134]    161.25    0.04   0.91   160.46   161.23   162.03   447 1.00
## mu[135]    152.66    0.29   2.59   151.64   152.99   153.56    80 1.04
## mu[136]    160.96    0.04   0.87   160.19   160.96   161.71   441 1.00
## mu[137]    148.60    0.26   3.04   147.47   148.97   149.59   136 1.04
## mu[138]    156.63    0.14   1.54   155.57   156.82   157.56   119 1.04
## mu[139]    138.85    0.19   2.43   137.86   139.04   140.00   164 1.04
## mu[140]    159.53    0.25   3.62   157.70   159.17   161.18   208 1.04
## mu[141]     87.59    0.06   1.41    86.58    87.67    88.70   538 1.01
## mu[142]    160.55    0.05   0.92   159.82   160.58   161.27   372 1.00
## mu[143]    149.50    0.26   2.96   148.41   149.87   150.47   134 1.04
## mu[144]    152.67    0.16   2.13   151.67   152.84   153.71   168 1.03
## mu[145]    118.01    0.22   2.77   116.79   118.33   119.00   164 1.04
## mu[146]    146.59    0.27   3.16   145.42   146.99   147.59   140 1.04
## mu[147]    152.32    0.29   2.64   151.30   152.65   153.23    85 1.04
## mu[148]    112.21    0.18   2.48   111.00   112.48   113.18   186 1.04
## mu[149]    108.20    0.10   1.40   107.14   108.23   109.21   188 1.02
## mu[150]    158.73    0.09   1.34   157.92   158.81   159.55   218 1.02
## mu[151]    156.00    0.18   1.80   155.08   156.22   156.88   105 1.04
## mu[152]    115.90    0.20   2.67   114.70   116.20   116.90   170 1.04
## mu[153]    114.28    0.12   1.61   113.32   114.36   115.32   175 1.03
## mu[154]    161.28    0.12   2.00   160.05   161.10   162.47   280 1.03
## mu[155]    140.89    0.28   3.34   139.65   141.29   141.94   145 1.04
## mu[156]    151.16    0.27   2.79   150.04   151.51   152.10   106 1.04
## mu[157]     92.76    0.08   1.53    91.78    92.87    93.80   367 1.02
## mu[158]    161.42    0.05   1.03   160.57   161.36   162.24   402 1.01
## mu[159]    160.94    0.04   0.87   160.17   160.95   161.69   439 1.00
## mu[160]    156.99    0.12   1.46   155.63   157.07   158.15   158 1.04
## mu[161]    161.45    0.10   1.68   160.35   161.31   162.52   298 1.03
## mu[162]    161.25    0.12   2.05   160.00   161.06   162.49   277 1.03
## mu[163]    152.42    0.29   2.62   151.41   152.75   153.32    84 1.04
## mu[164]    118.93    0.22   2.81   117.72   119.26   119.94   162 1.04
## mu[165]    151.22    0.27   2.79   150.10   151.57   152.15   105 1.04
## mu[166]    161.12    0.04   0.88   160.35   161.11   161.90   456 1.00
## mu[167]    154.06    0.29   2.33   153.06   154.34   154.93    65 1.04
## mu[168]    159.59    0.07   1.15   158.87   159.65   160.35   280 1.02
## mu[169]    152.17    0.28   2.66   151.13   152.50   153.08    88 1.04
## mu[170]    156.93    0.11   1.41   155.77   157.07   157.94   157 1.04
## mu[171]     91.09    0.07   1.48    90.10    91.19    92.14   408 1.01
## mu[172]     91.93    0.08   1.51    90.95    92.03    92.97   386 1.01
## mu[173]    153.90    0.29   2.36   152.88   154.19   154.78    65 1.04
## mu[174]    139.57    0.28   3.35   138.35   139.97   140.63   145 1.04
## mu[175]    160.88    0.16   2.51   159.49   160.64   162.26   237 1.03
## mu[176]    157.00    0.11   1.41   155.75   157.11   158.12   159 1.04
## mu[177]    146.06    0.27   3.19   144.87   146.47   147.08   140 1.04
## mu[178]    160.88    0.04   0.87   160.12   160.89   161.63   428 1.00
## mu[179]    160.81    0.17   2.58   159.40   160.57   162.20   234 1.04
## mu[180]    152.30    0.29   2.64   151.27   152.63   153.20    86 1.04
## mu[181]    161.24    0.04   0.91   160.46   161.22   162.02   449 1.00
## mu[182]    155.40    0.21   1.99   154.46   155.64   156.27    91 1.04
## mu[183]    158.19    0.32   4.46   156.05   157.74   160.11   199 1.04
## mu[184]    160.92    0.04   0.87   160.16   160.93   161.67   436 1.00
## mu[185]    142.91    0.19   2.45   141.95   143.12   144.07   165 1.04
## mu[186]    139.57    0.28   3.35   138.35   139.97   140.63   145 1.04
## mu[187]    122.33    0.24   2.96   121.12   122.67   123.36   156 1.04
## mu[188]    128.39    0.17   2.17   127.41   128.56   129.48   166 1.03
## mu[189]    121.29    0.15   1.90   120.28   121.43   122.38   169 1.03
## mu[190]    144.41    0.27   3.26   143.15   144.82   145.44   142 1.04
## mu[191]    159.42    0.07   1.19   158.69   159.48   160.18   270 1.02
## mu[192]    153.74    0.29   2.40   152.72   154.03   154.62    67 1.04
## mu[193]    154.85    0.25   2.15   153.84   155.12   155.71    74 1.04
## mu[194]    149.18    0.26   2.99   148.08   149.55   150.16   135 1.04
## mu[195]    151.44    0.27   2.76   150.33   151.78   152.37   101 1.04
## mu[196]     92.30    0.08   1.52    91.33    92.41    93.34   377 1.01
## mu[197]    154.66    0.26   2.19   153.70   154.93   155.53    71 1.04
## mu[198]    161.49    0.09   1.55   160.43   161.36   162.51   308 1.03
## mu[199]    156.99    0.11   1.40   155.75   157.11   158.06   158 1.04
## mu[200]    112.71    0.12   1.55   111.73   112.78   113.76   178 1.03
## mu[201]     98.84    0.09   1.29    97.80    98.80    99.85   223 1.02
## mu[202]    149.28    0.26   2.98   148.18   149.65   150.26   135 1.04
## mu[203]     94.86    0.09   1.36    93.78    94.79    95.88   231 1.02
## mu[204]    159.45    0.07   1.18   158.72   159.51   160.21   272 1.02
## mu[205]    150.68    0.27   2.85   149.56   151.03   151.62   115 1.04
## mu[206]    161.45    0.05   1.07   160.59   161.39   162.32   386 1.01
## mu[207]    148.79    0.26   3.02   147.67   149.17   149.78   136 1.04
## mu[208]    153.96    0.16   2.03   152.95   154.12   154.94   170 1.03
## mu[209]    154.06    0.29   2.33   153.06   154.34   154.93    65 1.04
## mu[210]    154.27    0.15   2.00   153.24   154.43   155.24   170 1.03
## mu[211]    137.59    0.28   3.35   136.38   137.99   138.66   146 1.04
## mu[212]    155.51    0.20   1.96   154.56   155.75   156.38    95 1.04
## mu[213]    119.29    0.22   2.83   118.08   119.61   120.29   161 1.04
## mu[214]    155.43    0.14   1.87   154.44   155.57   156.37   174 1.03
## mu[215]    161.24    0.04   0.91   160.46   161.22   162.02   449 1.00
## mu[216]    151.54    0.28   2.74   150.43   151.89   152.47    99 1.04
## mu[217]    110.52    0.11   1.47   109.51   110.57   111.54   182 1.02
## mu[218]    161.39    0.11   1.82   160.22   161.22   162.51   289 1.03
## mu[219]    154.50    0.27   2.23   153.50   154.77   155.36    68 1.04
## mu[220]    154.33    0.28   2.27   153.31   154.61   155.20    66 1.04
## mu[221]    100.91    0.09   1.28    99.89   100.90   101.93   215 1.02
## mu[222]    114.70    0.12   1.63   113.74   114.79   115.74   175 1.03
## mu[223]    149.65    0.18   2.31   148.64   149.83   150.75   166 1.04
## mu[224]    159.16    0.08   1.25   158.39   159.23   159.94   245 1.02
## mu[225]    160.62    0.05   0.91   159.88   160.64   161.34   383 1.00
## mu[226]    154.91    0.25   2.13   153.91   155.18   155.78    76 1.04
## mu[227]    119.45    0.14   1.82   118.46   119.58   120.55   170 1.03
## mu[228]    160.73    0.04   0.89   159.99   160.75   161.47   403 1.00
## mu[229]    151.46    0.28   2.75   150.36   151.81   152.39   100 1.04
## mu[230]    148.50    0.26   3.04   147.37   148.87   149.49   136 1.04
## mu[231]    152.32    0.29   2.64   151.30   152.65   153.23    85 1.04
## mu[232]    106.25    0.14   2.17   105.13   106.47   107.19   233 1.03
## mu[233]    148.29    0.26   3.06   147.16   148.67   149.29   137 1.04
## mu[234]    160.70    0.04   0.89   159.96   160.71   161.44   397 1.00
## mu[235]    149.09    0.26   3.00   147.97   149.46   150.07   135 1.04
## mu[236]    154.82    0.15   1.94   153.82   154.97   155.77   172 1.03
## mu[237]    156.73    0.13   1.50   155.64   156.90   157.65   130 1.04
## mu[238]    161.43    0.10   1.73   160.30   161.27   162.53   294 1.03
## mu[239]    154.87    0.25   2.14   153.86   155.14   155.73    75 1.04
## mu[240]    151.05    0.27   2.80   149.92   151.40   151.99   108 1.04
## mu[241]    129.92    0.27   3.22   128.64   130.29   130.99   148 1.04
## mu[242]    149.66    0.26   2.95   148.55   150.02   150.62   134 1.04
## mu[243]    114.11    0.19   2.58   112.87   114.40   115.10   177 1.04
## mu[244]    157.74    0.11   1.54   156.89   157.84   158.59   191 1.03
## mu[245]    138.99    0.28   3.35   137.77   139.39   140.05   145 1.04
## mu[246]    119.67    0.14   1.83   118.68   119.80   120.77   170 1.03
## mu[247]    159.30    0.08   1.22   158.56   159.36   160.07   259 1.02
## mu[248]    148.05    0.26   3.08   146.96   148.43   149.05   137 1.04
## mu[249]    101.30    0.12   1.91   100.20   101.50   102.25   267 1.03
## mu[250]    145.52    0.27   3.22   144.32   145.93   146.55   141 1.04
## mu[251]    151.16    0.27   2.79   150.04   151.51   152.10   106 1.04
## mu[252]    160.54    0.05   0.92   159.81   160.56   161.26   371 1.00
## mu[253]    160.46    0.05   0.94   159.73   160.49   161.18   359 1.01
## mu[254]    155.31    0.22   2.02   154.37   155.56   156.19    88 1.04
## mu[255]    155.96    0.14   1.81   154.99   156.09   156.88   176 1.03
## mu[256]    146.21    0.27   3.18   145.02   146.62   147.23   140 1.04
## mu[257]    157.65    0.11   1.55   156.80   157.75   158.51   190 1.03
## mu[258]    160.63    0.05   0.90   159.89   160.65   161.35   385 1.00
## mu[259]    146.36    0.27   3.18   145.18   146.77   147.37   140 1.04
## mu[260]    160.94    0.16   2.44   159.58   160.71   162.31   240 1.03
## mu[261]    148.83    0.26   3.02   147.70   149.20   149.81   136 1.04
## mu[262]    160.36    0.05   0.96   159.63   160.40   161.06   346 1.01
## mu[263]    145.68    0.27   3.21   144.48   146.09   146.70   141 1.04
## mu[264]     96.04    0.09   1.66    95.06    96.18    97.01   313 1.02
## mu[265]    151.54    0.28   2.74   150.43   151.89   152.47    99 1.04
## mu[266]     98.96    0.11   1.79    97.91    99.14    99.92   285 1.03
## mu[267]    154.35    0.28   2.27   153.34   154.63   155.22    66 1.04
## mu[268]    161.29    0.04   0.93   160.49   161.26   162.07   439 1.00
## mu[269]    161.27    0.12   2.03   160.03   161.08   162.50   279 1.03
## mu[270]    147.92    0.26   3.09   146.83   148.30   148.91   138 1.04
## mu[271]    149.21    0.18   2.33   148.20   149.39   150.33   166 1.04
## mu[272]    160.21    0.21   3.12   158.57   159.90   161.72   217 1.04
## mu[273]    116.70    0.21   2.71   115.49   117.01   117.70   167 1.04
## mu[274]    154.12    0.29   2.32   153.11   154.40   154.99    65 1.04
## mu[275]    158.62    0.09   1.37   157.81   158.70   159.45   213 1.02
## mu[276]    154.22    0.28   2.30   153.20   154.50   155.09    66 1.04
## mu[277]    158.77    0.09   1.33   157.96   158.85   159.59   219 1.02
## mu[278]    150.77    0.27   2.84   149.63   151.12   151.71   113 1.04
## mu[279]    153.88    0.29   2.37   152.86   154.17   154.76    66 1.04
## mu[280]    123.87    0.24   3.02   122.65   124.21   124.91   154 1.04
## mu[281]    143.11    0.28   3.30   141.87   143.52   144.16   143 1.04
## mu[282]    140.51    0.28   3.34   139.28   140.92   141.57   145 1.04
## mu[283]    145.33    0.27   3.22   144.11   145.74   146.36   141 1.04
## mu[284]    161.09    0.04   0.87   160.31   161.08   161.86   455 1.00
## mu[285]    161.51    0.08   1.44   160.50   161.40   162.49   321 1.02
## mu[286]    151.36    0.17   2.22   150.36   151.54   152.43   167 1.03
## mu[287]    150.80    0.27   2.83   149.67   151.15   151.74   113 1.04
## mu[288]    161.32    0.05   0.95   160.51   161.29   162.11   432 1.00
## mu[289]    158.32    0.10   1.43   157.50   158.40   159.15   203 1.03
## mu[290]    142.15    0.28   3.32   140.93   142.56   143.20   144 1.04
## mu[291]    116.41    0.21   2.69   115.20   116.72   117.41   168 1.04
## mu[292]    103.34    0.13   2.01   102.29   103.56   104.29   250 1.03
## mu[293]    161.17    0.04   0.89   160.40   161.16   161.94   455 1.00
## mu[294]    141.84    0.28   3.32   140.61   142.25   142.89   144 1.04
## mu[295]    158.23    0.10   1.44   157.41   158.32   159.07   201 1.03
## mu[296]    156.11    0.17   1.76   155.18   156.32   156.99   107 1.04
## mu[297]    152.85    0.29   2.56   151.81   153.18   153.75    77 1.04
## mu[298]    148.26    0.18   2.36   147.24   148.46   149.39   165 1.04
## mu[299]    123.14    0.24   2.99   121.92   123.48   124.17   155 1.04
## mu[300]    142.33    0.28   3.31   141.11   142.74   143.38   144 1.04
## mu[301]     88.83    0.06   1.43    87.83    88.91    89.92   484 1.01
## mu[302]    151.54    0.28   2.74   150.43   151.89   152.47    99 1.04
## mu[303]    154.49    0.15   1.98   153.47   154.64   155.45   171 1.03
## mu[304]    145.33    0.27   3.22   144.11   145.74   146.36   141 1.04
## mu[305]    161.37    0.05   0.99   160.55   161.33   162.17   417 1.01
## mu[306]    154.96    0.24   2.12   154.00   155.23   155.83    77 1.04
## mu[307]    125.82    0.25   3.09   124.58   126.18   126.87   151 1.04
## mu[308]     99.31    0.11   1.81    98.26    99.49   100.27   282 1.03
## mu[309]    156.47    0.15   1.61   155.49   156.67   157.38   112 1.04
## mu[310]    160.03    0.22   3.26   158.35   159.71   161.57   214 1.04
## mu[311]    153.42    0.29   2.46   152.40   153.73   154.31    70 1.04
## mu[312]    111.26    0.11   1.50   110.25   111.31   112.28   180 1.02
## mu[313]    161.49    0.06   1.16   160.61   161.42   162.39   362 1.02
## mu[314]    161.34    0.05   0.96   160.53   161.30   162.13   428 1.01
## mu[315]    142.85    0.28   3.30   141.62   143.26   143.90   144 1.04
## mu[316]    149.60    0.26   2.95   148.49   149.96   150.56   134 1.04
## mu[317]    159.64    0.24   3.54   157.85   159.29   161.26   209 1.04
## mu[318]    155.11    0.15   1.91   154.11   155.26   156.06   173 1.03
## mu[319]    161.50    0.06   1.20   160.60   161.43   162.42   353 1.02
## mu[320]    150.24    0.26   2.89   149.08   150.60   151.19   124 1.04
## mu[321]    158.30    0.10   1.43   157.48   158.38   159.13   203 1.03
## mu[322]    156.87    0.13   1.61   155.40   156.91   158.17   156 1.04
## mu[323]    154.55    0.27   2.22   153.58   154.83   155.42    69 1.04
## mu[324]     91.25    0.09   1.49    90.13    91.15    92.30   249 1.02
## mu[325]     92.30    0.09   1.45    91.21    92.21    93.34   242 1.02
## mu[326]     66.57    0.10   2.19    65.06    66.45    68.29   449 1.02
## mu[327]     93.45    0.09   1.41    92.37    93.36    94.48   235 1.02
## mu[328]    108.12    0.15   2.27   106.97   108.36   109.06   217 1.03
## mu[329]     86.44    0.06   1.40    85.41    86.51    87.58   594 1.01
## mu[330]    151.60    0.28   2.74   150.48   151.94   152.52    98 1.04
## mu[331]    102.07    0.12   1.95   100.98   102.27   103.02   260 1.03
## mu[332]     70.04    0.08   1.95    68.64    69.95    71.65   541 1.02
## mu[333]     92.69    0.09   1.43    91.60    92.60    93.73   240 1.02
## mu[334]     98.17    0.10   1.75    97.13    98.34    99.14   292 1.02
## mu[335]     85.75    0.12   1.78    84.52    85.58    86.99   216 1.02
## mu[336]    152.12    0.28   2.67   151.06   152.45   153.03    89 1.04
## mu[337]    157.18    0.12   1.63   156.29   157.30   158.06   184 1.03
## mu[338]    137.96    0.19   2.42   136.99   138.14   139.11   164 1.04
## mu[339]     91.19    0.07   1.48    90.20    91.28    92.23   405 1.01
## mu[340]    157.69    0.11   1.55   156.84   157.80   158.55   190 1.03
## mu[341]    151.11    0.27   2.80   149.98   151.46   152.04   107 1.04
## mu[342]    157.46    0.12   1.58   156.60   157.57   158.34   187 1.03
## mu[343]    155.22    0.22   2.05   154.25   155.47   156.10    84 1.04
## mu[344]    114.33    0.20   2.59   113.11   114.63   115.32   176 1.04
## mu[345]    100.96    0.12   1.89    99.86   101.15   101.90   270 1.03
## mu[346]    160.52    0.05   0.93   159.78   160.54   161.25   367 1.00
## mu[347]    152.66    0.29   2.59   151.64   152.99   153.56    80 1.04
## mu[348]    104.51    0.13   2.07   103.46   104.73   105.47   243 1.03
## mu[349]     97.00    0.09   1.31    95.93    96.95    98.02   226 1.02
## mu[350]    160.76    0.04   0.88   160.02   160.78   161.50   408 1.00
## mu[351]    159.55    0.07   1.16   158.82   159.60   160.31   277 1.02
## mu[352]    161.52    0.08   1.37   160.53   161.42   162.47   329 1.02
## mu[353]    160.40    0.05   0.95   159.67   160.43   161.09   351 1.01
## mu[354]     91.65    0.08   1.50    90.67    91.75    92.70   393 1.01
## mu[355]     70.21    0.20   2.96    68.53    69.89    71.93   209 1.03
## mu[356]    138.07    0.19   2.42   137.10   138.26   139.22   164 1.04
## mu[357]    153.48    0.29   2.45   152.47   153.79   154.37    69 1.04
## mu[358]    153.39    0.29   2.46   152.38   153.70   154.29    70 1.04
## mu[359]     91.44    0.09   1.48    90.33    91.34    92.50   248 1.02
## mu[360]     87.59    0.06   1.41    86.58    87.67    88.70   538 1.01
## mu[361]     77.32    0.19   2.38    75.85    77.08    78.91   155 1.03
## mu[362]     90.91    0.07   1.48    89.91    91.00    91.96   413 1.01
## mu[363]     71.53    0.07   1.85    70.17    71.47    73.11   611 1.01
## mu[364]     94.30    0.09   1.38    93.23    94.22    95.32   233 1.02
## mu[365]     73.75    0.07   1.72    72.42    73.71    75.24   700 1.01
## mu[366]    157.00    0.11   1.44   155.67   157.08   158.15   159 1.04
## mu[367]    154.29    0.28   2.28   153.27   154.57   155.16    66 1.04
## mu[368]     94.96    0.09   1.36    93.87    94.89    95.97   231 1.02
## mu[369]     95.42    0.09   1.35    94.33    95.36    96.44   229 1.02
## mu[370]     72.17    0.07   1.81    70.82    72.12    73.73   638 1.01
## mu[371]     78.91    0.05   1.49    77.70    78.92    80.25   872 1.01
## mu[372]    160.23    0.05   0.99   159.51   160.27   160.94   332 1.01
## mu[373]    133.00    0.27   3.30   131.75   133.38   134.07   147 1.04
## mu[374]    138.64    0.28   3.35   137.43   139.05   139.71   146 1.04
## mu[375]    160.97    0.15   2.40   159.62   160.75   162.33   242 1.03
## mu[376]    117.34    0.13   1.74   116.35   117.45   118.42   172 1.03
## mu[377]     80.87    0.17   2.11    79.50    80.67    82.28   163 1.03
## mu[378]     75.63    0.06   1.63    74.35    75.60    77.07   768 1.01
## mu[379]     87.31    0.06   1.40    86.28    87.38    88.41   551 1.01
## mu[380]    161.15    0.14   2.19   159.87   160.94   162.46   261 1.03
## mu[381]     74.38    0.06   1.69    73.07    74.35    75.85   724 1.01
## mu[382]    130.10    0.27   3.23   128.81   130.47   131.17   148 1.04
## mu[383]    160.36    0.05   0.96   159.63   160.40   161.06   346 1.01
## mu[384]     89.01    0.10   1.60    87.83    88.89    90.15   261 1.02
## mu[385]     95.13    0.09   1.62    94.17    95.27    96.13   325 1.02
## mu[386]    154.60    0.26   2.21   153.66   154.88   155.47    70 1.04
## mu[387]    154.83    0.25   2.15   153.83   155.10   155.69    74 1.04
## mu[388]    156.37    0.16   1.65   155.37   156.57   157.26   110 1.04
## mu[389]    159.93    0.06   1.06   159.20   159.98   160.66   307 1.01
## mu[390]    113.12    0.19   2.53   111.89   113.40   114.11   182 1.04
## mu[391]    112.15    0.11   1.53   111.15   112.21   113.19   179 1.03
## mu[392]     89.40    0.10   1.58    88.25    89.29    90.54   259 1.02
## mu[393]    153.74    0.29   2.40   152.72   154.03   154.62    67 1.04
## mu[394]    134.50    0.18   2.35   133.53   134.66   135.64   165 1.04
## mu[395]    114.70    0.12   1.63   113.74   114.79   115.74   175 1.03
## mu[396]     94.59    0.09   1.60    93.63    94.72    95.58   333 1.02
## mu[397]    161.09    0.14   2.27   159.78   160.87   162.41   253 1.03
## mu[398]    150.36    0.26   2.88   149.20   150.72   151.31   121 1.04
## mu[399]    133.23    0.18   2.32   132.26   133.41   134.37   165 1.04
## mu[400]    156.84    0.12   1.45   155.69   157.00   157.80   152 1.04
## mu[401]     83.43    0.14   1.93    82.14    83.24    84.73   185 1.03
## mu[402]     77.43    0.19   2.37    75.96    77.18    79.02   155 1.03
## mu[403]    159.78    0.06   1.10   159.05   159.84   160.52   294 1.01
## mu[404]    111.20    0.17   2.43   110.01   111.47   112.15   192 1.03
## mu[405]    152.40    0.17   2.15   151.39   152.57   153.44   168 1.03
## mu[406]    161.03    0.04   0.87   160.25   161.03   161.79   451 1.00
## mu[407]     71.96    0.07   1.83    70.60    71.90    73.52   630 1.01
## mu[408]    136.55    0.19   2.39   135.58   136.72   137.68   164 1.04
## mu[409]     78.37    0.18   2.30    76.93    78.14    79.90   157 1.03
## mu[410]    153.90    0.29   2.36   152.88   154.19   154.78    65 1.04
## mu[411]    161.39    0.05   1.00   160.56   161.35   162.19   411 1.01
## mu[412]    155.57    0.14   1.86   154.58   155.71   156.50   174 1.03
## mu[413]    147.46    0.26   3.12   146.36   147.85   148.46   138 1.04
## mu[414]    161.46    0.06   1.09   160.60   161.40   162.34   380 1.01
## mu[415]    154.39    0.28   2.26   153.37   154.66   155.26    67 1.04
## mu[416]     74.58    0.06   1.68    73.28    74.55    76.05   731 1.01
## mu[417]    153.37    0.29   2.47   152.36   153.68   154.27    70 1.04
## mu[418]    160.00    0.06   1.05   159.28   160.05   160.73   312 1.01
## mu[419]    149.78    0.25   2.94   148.67   150.15   150.74   133 1.04
## mu[420]    143.66    0.27   3.28   142.42   144.07   144.70   143 1.04
## mu[421]    124.51    0.16   2.03   123.53   124.66   125.60   167 1.03
## mu[422]    154.02    0.16   2.02   153.01   154.18   155.00   170 1.03
## mu[423]    147.32    0.27   3.12   146.21   147.70   148.31   139 1.04
## mu[424]    160.82    0.04   0.88   160.07   160.83   161.56   419 1.00
## mu[425]    144.85    0.27   3.24   143.61   145.26   145.89   142 1.04
## mu[426]     84.04    0.14   1.89    82.75    83.86    85.33   192 1.03
## mu[427]    158.13    0.10   1.47   157.31   158.22   158.96   199 1.03
## mu[428]    160.65    0.05   0.90   159.91   160.67   161.38   388 1.00
## mu[429]    139.87    0.19   2.44   138.89   140.07   141.04   164 1.04
## mu[430]    161.38    0.05   0.99   160.56   161.34   162.18   414 1.01
## mu[431]    153.82    0.29   2.38   152.79   154.11   154.70    66 1.04
## mu[432]    136.83    0.19   2.40   135.87   137.01   137.97   164 1.04
## mu[433]    118.25    0.14   1.77   117.26   118.37   119.34   171 1.03
## mu[434]    160.65    0.05   0.90   159.91   160.67   161.38   388 1.00
## mu[435]    153.89    0.16   2.03   152.89   154.06   154.88   170 1.03
## mu[436]    159.99    0.06   1.05   159.27   160.04   160.72   311 1.01
## mu[437]    160.98    0.15   2.39   159.63   160.76   162.34   243 1.03
## mu[438]    160.79    0.04   0.88   160.04   160.80   161.53   413 1.00
## mu[439]    156.14    0.17   1.74   155.21   156.35   157.03   107 1.04
## mu[440]     97.37    0.09   1.31    96.31    97.33    98.39   225 1.02
## mu[441]     69.50    0.09   1.98    68.10    69.41    71.13   522 1.02
## mu[442]    160.36    0.05   0.96   159.63   160.40   161.06   346 1.01
## mu[443]    144.77    0.27   3.25   143.53   145.18   145.81   142 1.04
## mu[444]    146.21    0.27   3.18   145.02   146.62   147.23   140 1.04
## mu[445]    156.40    0.16   1.64   155.39   156.59   157.29   111 1.04
## mu[446]    161.42    0.05   1.04   160.57   161.37   162.25   399 1.01
## mu[447]    159.90    0.06   1.07   159.17   159.95   160.63   304 1.01
## mu[448]    146.70    0.27   3.16   145.54   147.10   147.70   140 1.04
## mu[449]    130.02    0.17   2.22   129.05   130.19   131.14   165 1.03
## mu[450]    106.68    0.10   1.36   105.62   106.71   107.70   192 1.02
## mu[451]    160.81    0.04   0.88   160.06   160.82   161.55   417 1.00
## mu[452]    148.05    0.26   3.08   146.96   148.43   149.05   137 1.04
## mu[453]     76.35    0.06   1.59    75.08    76.33    77.78   801 1.01
## mu[454]    160.27    0.05   0.98   159.55   160.31   160.98   336 1.01
## mu[455]    132.43    0.27   3.28   131.16   132.81   133.50   147 1.04
## mu[456]     75.00    0.06   1.66    73.71    74.97    76.46   746 1.01
## mu[457]     76.05    0.19   2.48    74.55    75.79    77.65   163 1.03
## mu[458]    161.24    0.12   2.06   160.00   161.05   162.49   276 1.03
## mu[459]    144.56    0.19   2.44   143.60   144.76   145.71   165 1.04
## mu[460]    161.44    0.10   1.71   160.33   161.29   162.53   296 1.03
## mu[461]    160.75    0.04   0.88   160.01   160.77   161.49   406 1.00
## mu[462]    153.11    0.29   2.51   152.06   153.42   154.00    74 1.04
## mu[463]     71.74    0.07   1.84    70.38    71.69    73.32   621 1.01
## mu[464]    142.77    0.28   3.30   141.53   143.18   143.81   144 1.04
## mu[465]    156.67    0.14   1.52   155.60   156.85   157.59   123 1.04
## mu[466]    156.09    0.13   1.79   155.14   156.23   157.01   177 1.03
## mu[467]     62.48    0.13   2.51    60.81    62.31    64.38   378 1.02
## mu[468]     72.40    0.19   2.78    70.78    72.09    74.06   209 1.03
## mu[469]    154.06    0.29   2.33   153.06   154.34   154.93    65 1.04
## mu[470]    141.62    0.28   3.33   140.38   142.02   142.67   144 1.04
## mu[471]    104.01    0.13   2.05   102.96   104.23   104.96   246 1.03
## mu[472]     75.31    0.06   1.64    74.04    75.28    76.76   758 1.01
## mu[473]    160.27    0.05   0.98   159.55   160.31   160.98   336 1.01
## mu[474]    155.17    0.23   2.06   154.20   155.43   156.05    83 1.04
## mu[475]    113.28    0.19   2.53   112.05   113.56   114.26   181 1.04
## mu[476]     95.61    0.09   1.34    94.52    95.55    96.62   228 1.02
## mu[477]    160.32    0.05   0.97   159.60   160.36   161.02   342 1.01
## mu[478]    151.13    0.27   2.80   150.01   151.48   152.07   106 1.04
## mu[479]    161.27    0.04   0.92   160.48   161.25   162.05   442 1.00
## mu[480]    154.59    0.27   2.21   153.63   154.86   155.46    69 1.04
## mu[481]     87.31    0.06   1.40    86.28    87.38    88.41   551 1.01
## mu[482]    156.94    0.12   1.67   156.03   157.06   157.83   182 1.03
## mu[483]    151.89    0.28   2.70   150.78   152.23   152.81    93 1.04
## mu[484]     83.23    0.14   1.94    81.93    83.03    84.53   183 1.03
## mu[485]    159.16    0.08   1.25   158.39   159.23   159.94   245 1.02
## mu[486]    156.90    0.11   1.42   155.76   157.05   157.89   157 1.04
## mu[487]    120.49    0.14   1.87   119.47   120.62   121.58   170 1.03
## mu[488]     91.73    0.09   1.47    90.62    91.63    92.78   246 1.02
## mu[489]    161.01    0.04   0.87   160.23   161.01   161.76   448 1.00
## mu[490]    131.62    0.27   3.27   130.34   132.00   132.70   148 1.04
## mu[491]    161.42    0.05   1.03   160.57   161.36   162.24   402 1.01
## mu[492]    153.71    0.29   2.40   152.71   154.01   154.60    67 1.04
## mu[493]    110.85    0.11   1.48   109.83   110.90   111.87   181 1.02
## mu[494]    144.73    0.27   3.25   143.49   145.14   145.77   142 1.04
## mu[495]    142.85    0.28   3.30   141.62   143.26   143.90   144 1.04
## mu[496]    152.96    0.29   2.54   151.90   153.28   153.86    76 1.04
## mu[497]    140.14    0.28   3.34   138.90   140.54   141.20   145 1.04
## mu[498]    160.03    0.06   1.04   159.31   160.08   160.76   314 1.01
## mu[499]    156.61    0.13   1.72   155.69   156.74   157.51   180 1.03
## mu[500]    152.76    0.16   2.13   151.76   152.93   153.79   168 1.03
## mu[501]    157.81    0.11   1.52   156.97   157.92   158.66   192 1.03
## mu[502]    161.49    0.09   1.55   160.43   161.36   162.51   308 1.03
## mu[503]    153.04    0.29   2.52   151.97   153.36   153.93    75 1.04
## mu[504]    121.37    0.15   1.90   120.35   121.51   122.45   169 1.03
## mu[505]     86.63    0.06   1.40    85.61    86.70    87.76   584 1.01
## mu[506]    154.55    0.15   1.97   153.54   154.70   155.51   171 1.03
## mu[507]    153.33    0.29   2.47   152.31   153.64   154.22    71 1.04
## mu[508]     99.75    0.11   1.83    98.69    99.93   100.70   279 1.03
## mu[509]    122.31    0.15   1.94   121.30   122.46   123.39   168 1.03
## mu[510]    161.52    0.07   1.31   160.56   161.43   162.45   336 1.02
## mu[511]    148.53    0.26   3.04   147.40   148.90   149.52   136 1.04
## mu[512]    110.81    0.17   2.41   109.61   111.08   111.76   195 1.03
## mu[513]     90.09    0.10   1.54    88.95    89.98    91.18   256 1.02
## mu[514]     93.50    0.08   1.56    92.54    93.61    94.50   352 1.02
## mu[515]    152.90    0.29   2.55   151.85   153.22   153.80    77 1.04
## mu[516]    145.25    0.27   3.23   144.03   145.66   146.28   141 1.04
## mu[517]    153.98    0.29   2.35   152.97   154.27   154.86    65 1.04
## mu[518]     60.12    0.14   2.70    58.35    59.93    62.10   348 1.02
## mu[519]    148.09    0.26   3.07   146.99   148.47   149.08   137 1.04
## mu[520]     63.71    0.12   2.41    62.08    63.55    65.52   397 1.02
## mu[521]    100.64    0.09   1.28    99.63   100.63   101.67   216 1.02
## mu[522]    125.55    0.16   2.07   124.57   125.71   126.64   166 1.03
## mu[523]     84.00    0.05   1.40    82.95    84.05    85.19   724 1.00
## mu[524]    158.42    0.10   1.41   157.61   158.50   159.25   206 1.02
## mu[525]    153.84    0.29   2.38   152.81   154.13   154.72    66 1.04
## mu[526]    129.70    0.17   2.21   128.72   129.87   130.81   165 1.03
## mu[527]    119.50    0.22   2.84   118.29   119.82   120.51   161 1.04
## mu[528]     96.21    0.09   1.67    95.23    96.36    97.19   311 1.02
## mu[529]    160.53    0.05   0.92   159.79   160.55   161.25   369 1.00
## mu[530]    149.63    0.26   2.95   148.52   149.99   150.59   134 1.04
## mu[531]    124.72    0.25   3.05   123.50   125.07   125.77   153 1.04
## mu[532]     84.00    0.05   1.40    82.95    84.05    85.19   724 1.00
## mu[533]    155.85    0.14   1.82   154.88   155.99   156.77   175 1.03
## mu[534]    151.11    0.27   2.80   149.98   151.46   152.04   107 1.04
## mu[535]    155.63    0.19   1.92   154.70   155.87   156.50    99 1.04
## mu[536]    110.36    0.11   1.47   109.34   110.41   111.37   182 1.02
## mu[537]     71.64    0.07   1.85    70.28    71.58    73.22   616 1.01
## mu[538]    142.85    0.28   3.30   141.62   143.26   143.90   144 1.04
## mu[539]     79.10    0.18   2.24    77.68    78.88    80.63   158 1.03
## mu[540]    142.21    0.19   2.45   141.25   142.42   143.37   164 1.04
## mu[541]    161.50    0.06   1.20   160.60   161.43   162.42   353 1.02
## mu[542]    156.94    0.12   1.53   155.51   157.00   158.17   157 1.04
## mu[543]     79.21    0.18   2.23    77.78    78.99    80.74   158 1.03
## mu[544]    161.52    0.07   1.28   160.58   161.43   162.44   340 1.02
## lp__     -1323.61  164.14 238.94 -1685.71 -1190.02 -1188.07     2 4.82
## 
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:59:50 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).